Is special relativity axiomatic? Can special relativity be somewhat described by the following statement?

If the speed of light is a universal constant, and if light can be placed in separate frames of motion, then time must be altered by speed.

 A: Asking whether it's axiomatic is different from asking whether a particular axiomatization works. Some examples:


*

*The original axiomatization in Einstein's 1905 paper involved two postulates.

*Sometimes people telescope Einstein's two postulates into one, saying that all the laws of physics (both mechanical and optical) are the same in all frames.

*The question proposes a one-postulate axiomatization, but I don't think it works, because it never connects to the mechanical laws. If you don't talk about mechanics at all, then you can't describe clocks and rulers, and then I don't think you've got a complete theory of SR.

*Ignatowsky 1911, shortly after Einstein, did an axiomatization in terms of symmetry, which is much more modern than Einstein's approach.

*One can also take an approach in terms of maximizing proper time for test particles (Laurent 1994).
IMO Einstein's original axiomatization deserves to be forgotten, since it's clearly inappropriate from the modern point of view. Today, we don't think of light as playing any special role in relativity or in physics in general.
W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972; cf. Palash B. Pal, "Nothing but Relativity," Eur.J.Phys.24:315-319,2003, http://arxiv.org/abs/physics/0302045v1
Bertel Laurent, Introduction to spacetime: a first course on relativity
A: Let's have your statements. 
"...If the speed of light is a universal constant, and if light can be placed in separate frames of motion..."
If I understood you correclty, that means, that the speed of light is constant relative to every inertial frames. If yes, your statement really leads dependence of time on the speed in different frames. Let's have one-dimensional motion: if $c = \frac{dx}{dt} = const$, for new frame which have an arbitrary (!) speed $u$ relative to an old frame, you will have
$$
c' = \frac{dx'(u, x, t)}{dt'} = c. 
$$
But it is possible only in case $dx' = cdt'$ (only in case when speed which we analyzing is equal to $c$ (!)), so $dt' = f(u, x, t)$ in general.
Also I can add some info about axiomatization of special relativity and role of speed of light in SR formalism. 
The special relativity's kinematics can be derived by the following axioms (which can be, more or less, proved by the experiments):


*

*Homogeneity of the space-time (energy and impulse conservation).

*Relativity principle (compairing of some processes depends on the different frames).

*Isotropy of space (classical angular momentum conservation).

*The principle of causality (for example, some statistical physics and thermodynamics arguments).


These principles leads us to the Lorentz transformations, to the invariance of some fundamental speed $c$ under these transformations (in the general case it may not match the speed of light) and to the maximum "speed of events" limited by $c$. Then, we can experimentally show, that speed of light is also invariant under choosing of inetrial frames, so in this case we can identify it with $c$. So it means that even if the speed of light was not invariant, the special theory of relativity would be true.
By the way, by adding an axiom of absoluteness of simultaneity we can get Galilean space and corresponding transformations.
A: 
Is special relativity axiomatic?

Yes, in the following sense:
in the theory of relativity there can be certain foundational or "primitive" notions identified which are considered universally self-evident and comprehensible without need of any further definition, but from which definitions of all other notions are constructed. 
As Einstein put it:
All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more [...] material points.
Some details on how to use the notions of "material point(s)" and "coincidence" to construct definitions of terminology such as "mutual rest", "duration", "distance", "speed", "inertial motion" and so on may be found (for instance) in
John W. Schutz, "Independent Axioms for Minkowski Space-Time", (Chapman & Hall/CRC Research Notes in Mathematics. Book 373, 1997).

If the speed of light is a universal constant

... which is of course a consequence of the (universally applicable) definition of (how to measure) "speed" (and "refractive index") in the first place ...

and if light can be placed in separate frames of motion, then time must be altered by speed.

This part of your statement seems quite incomprehensible to me; perhaps you should try to rephrase what you mean more explicitly in terms of the "primitive" notions "material point(s)" and "coincidence" ...
A: Yes, more or less. In physics one usually speaks of postulates, rather than axioms, but the idea is the same: a starting point is posed (in physics these would follow from experiment), from which the properties of the theory are obtained by deduction. 
In wikipedia the postulates of special relativity are formulated as


*

*First postulate (principle of relativity)
The laws of physics are the same in all inertial frames of reference.

*Second postulate (invariance of c)
The speed of light in free space has the same value c in all inertial frames of reference.
In the same article there also is a precise mathematical formulation.
This should be sufficient to derive time dilation and Lorentz contraction, but to derive something like $E = mc^2$ some physics is needed (they cannot be obtained by pure deduction from the postulates).
EDIT
I'd like to add a clarification (in reaction to some of the comments): the first postulate states that the laws of physics are the same in all inertial frames. It is not necessary to assume any particular law of physics. In particular nothing is said in the postulates about Newton's laws or Maxwell's equations. From the postulates in this form it already follows that the light cone is preserved in all frames of reference. It is a theorem of Zeeman that this implies that the frames are related by the composition of a translation, a dilation and a Lorentz transformation. As is well known, time dilation and Lorentz contraction easily follow from properties of Lorentz transformations. 
A: Yes and No. Yes because it can be "postulated" and No because, apart from all experimental subtleties of the notions involved, there are also inequalities outlining the region of its applicability. Point-likeness, classicality, etc. - all are approximations with the corresponding inequalities. Inequalities are an essential part of any branch of theoretical physics.
