I know that Special Relativity is a classical theory, so perhaps it applies to light waves, but I'm not too sure. The question I want to ask is, does Special Relativity set a bound on particle velocity, wave velocity, or both? And why?
It applies to both in a number of ways.
In QFT: There isn't so much talk of waves moving in time in QFT, because scattering results which comprise most QFT results are in the $t \to \infty$ limit. The closest thing I have to offer here when it comes to either a particle or a wave moving at at most $c$, is that momentum eigenstates transform via lorentz transformation matrices on the momentum 4-vector. This naturally limits the corresponding velocity never to exceed $c$ in another frame.
In terms of actual particles or waves: When the Dirac equation is used in relativistic quantum mechanics as a wave function, the wave cannot spread faster than $c$. That means if it is originally $0$ except in some finite region, all nonzero points can only have a finite wavefunction value once light has had time to reach that point from the nonzero region. This can be seen as a corollary of the fact that if Bohmian Mechanics is used to track trajectories with the dirac equation, then those trajectories do not move faster than $c$. So there we have both particle and wave. Whether you accept Bohmian trajectories as the actual particle trajectories, and whether you believe actual trajectories exist, is of course up to you and a completely unanswered question.
I said that this covers both particle and wave, but this discussion would be incomplete without mentioning that one thing that could potentially get in the way is wavefunction collapse. If you believe that a momentum measurement collapses a wavefunction to a perfect momentum eigenstate, then the particle could be anywhere in the world with equal probability on a subsequent measurement, obviously exceeding the light speed limit. But I think the measurement postulates are artificial anyway, so this isn't really something physical. Many physicists don't tend to take those postulates too literally, but now I guess it is become time for this to be admitted in textbooks too, when the postulates are introduced.