Would it be possible to develop special relativity without knowing about light? My understanding of special relativity is that it is fundamentally based on the constancy of the speed of electromagnetic radiation - that this speed is a physical law (or derivable from physical laws and facts about the vacuum)  and not a contingent fact. Does that mean that, if humanity did not know about light (or electromagnetic radiation, generally), would we then be unable to come up with special relativity? To put it another way, imagine a species of intelligent life insensitive to electromagnetic radiation - no eyes etc. Would they still have a reason to produce the theory of relativity, or would they likely have created a physics framework which doesn't involve the speed of light so centrally? 
 A: N. David Mermin, Relativity without light.
Abstract:
The relativistic addition law for parallel velocities is derived directly from the principle of relativity and a few simple assumptions of smoothness and symmetry, without making use of the principle of the constancy of the velocity of light.
There is also a chapter devoted to this topic in N. David Mermin, Boojums All the Way through
Communicating Science in a Prosaic Age:
Note that I do not have access to either of those sources, although the OP may do, or he/she may wish to obtain such access.
Here is a free article Relativity without light: a further suggestion
Shan Gao on the related subject and its abstract:
The role of the light postulate in special relativity is reexamined. The existing theory of relativity without light shows that one can deduce Lorentz-like transformations with an undetermined invariant speed, based on homogeneity of space and time, isotropy of space and the principle of relativity...
Another free article, this one courtesy of CERN:  http://cdsweb.cern.ch/record/940058/
and its abstract:
Using only the reciprocity postulate, it is demonstrated that the spatial separation of two objects, at rest in some inertial frame, is invariant. This result holds in both Galilean and special relativity. A corollary is that there are no `length contraction' or associated 'relativity of simultaneity' effects in the latter theory. A thought experiment employing four unsynchronised clocks and a single measuring rod provides a demonstration of the time dilatation effect. This effect, which is universal for all synchronised clocks at rest in any inertial frame, is the unique space-time phenomenon discriminating Galilean and special relativity.
A: Purely empirically, we have various atomic clock experiments that clearly demonstrate relativistic effects. There have been mountain-valley experiments demonstrating gravitational time dilation (Briatore 1979), experiments capable of separately detecting kinematic and gravitational time dilation (Chou 2010), as well as the famous Hafele-Keating experiment, which was sensitive to a mixture of gravitational time dilation, kinematic time dilation, and the Sagnac effect.
From the theoretical side, light plays no important role in the modern understanding of relativity, and the $c$ in all the relativistic equations is not really interpreted as the speed of light but as a conversion factor between time and space. It was realized very early on (Ignatowsky 1911) that special relativity could be developed axiomatically starting from assumptions about nothing more than symmetry principles such as homogeneity and isotropy of space. For modern, elementary presentations in this style, see Pal 2003 and this one that I wrote.
L. Briatore and S. Leschiutta, Evidence for the earth gravitational shift by direct atomic-time-scale comparison, Il Nuovo Cimento B, 37B (2): 219 (1979), http://www.scribd.com/doc/106593804/briatore-1977 . A very cool similar experiment by an amateur experimenter is described here: http://leapsecond.com/great2005/
Chou, http://www.sciencemag.org/content/329/5999/1630.abstract ; described at http://www.scientificamerican.com/article.cfm?id=time-dilation
W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972. A more recent exposition that's free online is Palash B. Pal, "Nothing but Relativity," (2003) arxiv.org/abs/physics/0302045v1
Pal, http://arxiv.org/abs/physics/0302045
A: I have no idea whether scientists would have imagined relativity from
theoretical considerations alone.  But I do know that if they had not
developed it a century ago, they would be doing so now, with the tax
payer on their back. The tax payer would be eager to know why they are
not capable to have decent clocks to keep accurate time on bord the
GPS satellites, after spending billions to put them in place.
Now this does not exactly answer the question since there would be no
GPS without radio communication, hence electromagnetic waves. But I
guess it is still in the spirit of the question as light does not play
an essential role in the problem they would encounter.

Since I am downvoted, I will try to explain further. I would
appreciate the downvoter to explain why he does not agree with my initial
answer. At least that might teach me something
(and possibly others too).
In most cases, scientists work on a problem, on a new theory, because
there is a phenomenon they cannot explain, which was apparently the case for special relativity. Possibly it can also be to unify existing knowledge, which seems to be the case for general relativity.
So I wondered what would motivate people who do not experiment with
light, or who do not know enough about it to be able to measure its speed.
Another major phenomenon due to relativity is differences in local time.
My point is that the GPS depends on extremely accurate timing, so as
to have accurate knowledge of the positioning of the satellites, in order to give
precise positioning on earth. And both SR and GR affect the time on
board satellites (see Time_dilation in
wikipedia). http://en.wikipedia.org/wiki/Time_dilation I once looked
at the calculation, but all I remember is that without relativistic
correction, the time variation will amount to errors of several meters
per year (maybe per month). And it could be worse if SR and GR did not
have opposite effects.
For people who do not know about time dilatation, the effect of
relativity on satellite clocks would be seen as malfunction. (I read that this is precisely what the CMB looked like at first: antenna malfunction)
Now, if the use of radio for communication must be excluded. There
have been several experiments that measure time dilatation on earth,
as noted by @Ben Crowell.
Taking a clock in a satellite and bringing it back works too. That is
probably what would be tried with a copy of the "malfunctioning" GPS
clock. The interest of the GPS case is that it is now considered
a common appliance by most people, who are still surprised to learn that
its functionning involves relativity in a significant way.
But saying relativity can be derived from theoretical considerations
alone is not an answer if you will not provide experiments that will
show that it does make a difference. And if you do provide such
experiments, then they might as well be the original discrepency
between experience and theory that motivate developing a new theory.
Alternatively the new theory should at least be a unification of existing theories. But which should they be in the case of SR (I may have missed the point of some technical explanations of previous posts).
Another point is that Einstein thinking about the subject was
apparently also conditioned by his work on time at the Swiss patent
office.  I read that clock synchronization for railways stations was then
an important topic, and the subject of many patent
applications. Railways were still new. And Eistein was allegedly the expert on such paptents.
Finally, an interesting development on this is the scientific work of Leslie
Lamport in distributed computer systems (see the web). Synchronizing
clocks and dating events is important in a variety of
applications of distributed computing. The specialists had developed various (equivalent)
axiomatizations of the problem, but lacking fundamental understanding
of their justification.  Lamport justified these axioms by a physical
analysis of the problem which amounts to a qualitative version of the
theory of relativity, to which he explicitly referred.
So, it is possible that if the physicists had not developed the theory
of relativity, the computer scientists might have done it to some extent, at least
qualitatively.
Indeed, if people did not have electromagnetic waves, they would have
no radio, and thus they would rely a lot on physical transportation
which would increase the importance of clock synchronization issues.
Which brings us back to Einstein's work at the Swiss patent office.
A: I'd like to add to the answers of @BenCrowell, @DavidZaslavsky and @Mitchell. It's easy to get stuck on $c$ being the speed of light, since that is how the popular press define this constant in the iconic $E=m\,c^2$, and this leads everyone to wonder "what has light in particular got to do with it?". It is more helpful to think of $c$ as the freespace speed of a massless particle (zero rest mass) - a universal property of the World independent of the particular phenomenon (light in this case). It might also be helpful to imagine the following alternative possible history of physics, wherein special relativity comes before the Michelson-Morley experiment.
In this imaginary history, an Einstein-like character might well have begun by deeply questioning Galileo's relativity, wherein we add velocity vectors in the "normal", vector algebra way to work out transformation the of observed velocities, times and distances that happen when we shift from one inertial frame of reference to another. As well as adding vectors according to the parallelogram law in Galilean relativity, times and distances are the same for all inertial observers. Let's imagine our OWE (other world Einstein) asking the probing question - "is this the only logical way it could be?".
So then our OWE might make some VERY basic and reasonable assumptions about symmetry: if I transform from frame (1) to a frame (2) moving at a constant speed $v_{1,2}$ in some direction then my distance and time co-ordinates are transformed by some matrix $T(v_{1,2})$ (OWE is trying to find the most general form of the matrix function of velocity $T(v)$). If I then transform to a third frame (3), one moving at velocity $v_{2,3}$ in the same (original) direction relative to the transformed frame (2) (using the matrix $T(v_{2,3})$, this has to be equivalent to a single transformation $T(v_{1,3})$ from the first to the third frame with some relative velocity $v_{1,3}$. In particular, if frame (3) is moving relative to frame (2) at velocity $-v$, then frames (1) and (3) have to be the same and $T(v) T(-v) = I$ (here $I$ = identity transformation - my running away from you at velocity $v$ should seem the same as your running away from me at the same speed in the opposite direction). This symmetry arises from a basic "homogeneity" (space and time are the "same" in some sense everywhere) and the Copernican notion that there is no special frame. Notice how one has NOT assumed that $v_{1,2}+v_{2,3} = v_{1,3}$, aside from in the special case of when $v_{1,2} = -v_{2,3}$. 
Now, if our OWE makes these assumptions ALONE (see the  Wikipedia entry  (cf Ignatowsky 1911 cited by @BenCrowell) he or she would find that not only is Galilean relativity a possibility, but that there is a whole class of relativities possible, with one unspecified velocity parameter $c$: their transformation equations all have the same form but the $c$ parameter is different. Galilean relativity is the relativity that one gets as the $c$ parameter goes to infinity.
So our OWE says to themself: "we've been assuming Galilean relativity all along, but what if the $c$ parameter for our World is just very big, not infinite, so that we've hitherto not measured it as finite. What experiments could we do to find out what $c$ is?". After some work with a relativity with a finite $c$, OWE would deduce that:


*

*$c$ would be measured to be the same in all inertial reference frames. Moreover, so as to enforce this invariance of $c$, there would be a peculiar addition rule for velocities not quite the same as the parallelogram rule;

*No material object can go faster than $c$ and indeed something can travel at speed $c$ only if it has a rest mass of zero.


So now along come Michelson and Morley with their famous experiment. As it happenned, their experiment implied that the speed of light is the same in all inertial frames. The addition rule for velocities is the same as OWE's weird equations if we take $c$ to be the speed of light.
So now, in this other World, the Michleson-Morley experiment has the following interpretation: Galilean relativity is almost certainly wrong, because we've found something that behaves just like OWE's predictions assuming a non-infinite $c$. Moreover, light MUST be a massless (zero rest mass) thing.
Notice how too if a positive result would have been gotten from the MM experiment, this would ALSO fit in with OWEs thinking. Suppose the MM experiment showed a delay difference of a few fringes depending on the interferometer's direction. Suppose there were only a very few fringes - definitely not a null result, but way too few fringes to be consistent with Galilean addition of aether and Earth velocities. Then this would also support OWE - the failure of Galilean addition would show that there is probably a finite value of $c$, but the fact that the result was not null would show that the speed of light were not quite as great as the universal speed limit $c$. Moreover, the MM experiment in this World would conclude light has a rest mass, which one could estimate from the experimental result and moreover one could deduce a good estimate for the value of $c$ as well.
A: 
My understanding of special relativity is that it is fundamentally based on the constancy of the speed of electromagnetic radiation

Actually, no, that's true. Special relativity is (one could say) based on the existence of some speed that is invariant in all inertial reference frames. The fact that light (or anything else) travels at that speed is pretty much irrelevant. So while the development of SR was originally prompted by the constancy of the speed of light, it's not a requirement to have developed the theory. The effects of relativity would become apparent just from the time dilation/length contraction experienced by any sufficiently fast-moving object.
A: It is, in fact, possible to derive the mathematics of the theory of relativity without reference to light. David Mermin did so in 1984 (Mermin N D 1984 Relativity without light Am. J. Phys. 52.2 119–24). According to the abstract, "The relativistic addition law for parallel velocities is derived directly from the principle of relativity and a few simple assumptions of smoothness and symmetry, without making use of the principle of the constancy of the velocity of light."
So, being insensitive to electromagnetic radiation does not preclude the derivation of relativity, although the lack of a speed-of-light-centered theory (depending on how you define this) is still an open question. c might just mean something conceptually different to them, but it'll still show up.
A: Albert Einstein was largely motivated by Maxwell's equations and the electromotive force in particular.  So it would be quite possible to develop special relativity without the special case of light.  Amphere's law is a relativistic effect, so that would be enough.
A: No, it wouldn't be possible to develop special relativity without light. Scientists developed a model of the propagation of light in a material substance called the ether based upon what they knew about the propagation of sound waves in a medium. Their expectation was they would be able to detect the speed at which the Earth travelled through it, but experiments gave a null result which scientists found initially confusing. Eventually this gave birth to the Special Theory of Relativity.
You therefore need to know about light behaving differently to what you know about sound, to see there is something interesting going on.
A: What if Bats indeed would have been intelligent? :) 
Think about it like that:
Any changes requires signal trasmission of ANY kind (in bats case sound in the air). 
There are 2 cases:
1) Signal trasmission is infinitely fast - in this case we get Galileo-Newton mechanics.
2) Signal trasmission is finite - in this case we have SR.
What is signal? Signal is a general process(phenomenon) by which far world ALIENS determine such conceptions as SPEED, TIME and so on (we use light for everything). If they dont have these conceptions then we are in trouble and we can not asnwer your question casue WE DONT KNOW, but if there is some sort of speed and time variables in their physics picture then they will rediscover special relativity.
