Is the speed of gravitational waves constant in all inertial frames? If so, why isn't this included as one of the postulates of special relativity?
The light speed postulate could have referred to anything applicable, or simply said there is an invariant speed. This implies Lorentz invariance instead of Galilean invariance. The resulting energy-momentum relation includes a parameter called the rest mass or invariant mass. We can then prove traveling at speed $c$ is equivalent to this parameter being $0$, which applies not only to photons of light but also to any massless particle or field.
Your question is a bit more involved than one might think because it depends on exactly what you mean by a gravitational wave.
In relativity, both special and general, the local speed of a massless particle is always equal to the speed of light. Note that I say local speed because the speed of light is complicated in general relativity. For more on this see GR. Einstein's 1911 Paper: On the Influence of Gravitation on the Propagation of Light.
Now, when we say gravitational waves we normally mean an infinite plane wave of a magnitude small enough that it's propagation is linear. By this we mean that the energy if the gravitational wave is low enough that we can neglect the effect the energy of the wave has on the geometry of spacetime. In this case the local speed of the gravitational wave is always the same as the local speed of light. We don't include this as a postulate because it isn't a postulate. We get this for free from Einstein's equations.
When we get spacetime fluctuations so intense that we can no longer ignore backreaction life gets more complicated and I confess I'm not sure what the answer is. Part of the problem is that speed is not well defined for something that isn't an infinite plane wave. However my understanding is that the speed at which all changes to the spacetime geometry propagate is basically the speed of light.