Before Maxwell's equations, physics postulated what is now called Galilean relativity. In a vacuum, electromagnetic waves obeying Maxwell's equations propagate at speed $c:=\left(\mu_0\varepsilon_0\right)^{-1/2}$. Special relativity came from the realisation that the transformation between coordinate systems couldn't be Galilean if all observers are to agree on physical laws. With some group theory postulates, we can show physics will either obey Galilean or special relativity. The two cases differ depending on whether a parameter in the calculation with units of $\text{velocity}^{-2}$ is $0$ or positive. The former case is Galilean; the latter gives a speed that is invariant under the addition of velocities. Galilean physics has no finite speed with this property, whereas special relativity doesn't allow multiple invariant speeds.
A massless field will satisfy a wave equation with some speed. (To call a field massless is a comment on what its equation of motion looks like; it doesn't commit us to a belief in a particle counterpart with mass $0$, although if it does have such a counterpart then quantum mechanics implies that mass inference is correct.) For such a physical law to be invariant, its wave speed must be too. However, only one invariant wave speed is allowed for. In fact, once special relativity emerged as a necessary consequence of classical electromagnetism, it was realised that said electromagnetism, while already consistent with special relativity, could be written in a new way that makes this fact obvious even if we take special relativity as axiomatic first. (This approach defines $c$ as the invariant speed, rather than in terms of $\mu_0,\,\varepsilon_0$.) To do this, we write Maxwell's equations in terms of the electromagnetic tensor $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$, where $E_i=cF_{0i},\,B_i=-\frac{1}{2}\sum_{jk}\epsilon_{ijk}F_{jk}$. One resulting equation is $\partial_\mu F^{\mu\nu}=\mu_0 J^\nu$; note that this provides an alternative definition of $\mu_0$, since in this approach it is less fundamental than $c$. This shows that $c$ isn't an artefact of electromagnetism per se; $\mu_0$ is.
Gravitational waves aren't perfectly analogous to electromagnetic ones. Four-momentum conservation limits them to quadrupole terms, which weakens the signal so much we've only just developed technology that can detect them. Controlling them to send messages is a long way off, but it's in principle possible. But the speed-$c$ results in electromagnetism are derived by considering waves without assuming a particulate quantisation of them (although quantum mechanics tells us such a quantisation does in fact exist). The speed of gravitational waves can be proven to be $c$ without introducing gravitons explicitly in one's assumptions.
