Further to Timaeus's Answer, the second postulate follows from the first postulate if we know about light. Otherwise, the second cannot follow from the first in a strict sense.
However, even if you don't know about light, there is still a way whereby the second postulate can be strongly motivated by the first, as follows.
The first postulate is essentially Galileo's notion of relativity as explained by his Allegory of Salviati's Ship.
If you assume:
- The first relativity postulate; and
- A concept of absolute time, i.e. that the time delay between two events will be measure to be the same for all inertial observers; and
- Homogeneity of space and time so that linear transformation laws between inertial frames are implied (see footnote)
Then these three assumptions alone uniquely define Galilean Relativity.
However, if you ask yourself "what happens to Galileo's relativity if we relax the assumption of absolute time" but we keep 1. and 3. above, then instead we find that a whole family of Lorentz transformations, each parametrised by a parameter $c$, are possible. Galilean relativity is the limiting member of this family as $c\to\infty$. The study of this question was essentially Einstein's contribution to special relativity. You can think of it as Galileo's relativity with the added possibility of an observer-dependent time. I say more about this approach to special relativity in my answer to the Physics SE Question "What's so special about the speed of light?".
It follows from this analysis that if our Universe has a finite value of $c$, then something moving at this speed will be measured to have this speed by all inertial observers. However, there is nothing in the above argument to suggest that there actually is something that moves at this speed, although we could still measure $c$ if we can have two inertial frames moving relative to each other at an appreciable fraction of $c$. It becomes a purely experimental question as to whether there is anything whose speed transforms in this striking way.
Of course, the Michelson Morley experiment did find something with this striking transformation law.
Footnote: The homogeneity of space postulate implies the transformations act linearly on spacetime co-ordinates, as discussed by Joshphysic's answer to the Physics SE question "Homogeneity of space implies linearity of Lorentz transformations". Another beautiful write-up of the fact of linearity's following from homogeneity assumptions is Mark H's answer to the Physics SE question "Why do we write the lengths in the following way? Question about Lorentz transformation".