The Pauli Exclusion Principle comes from the fact that the wave functions of particles with half integer spin are antisymmetric under particle exchange. From how I understand it, this relationship arises from combining relativity with quantum mechanics. Does the Pauli Exclusion Principle still hold if $c=\infty$ or does it require a finite value for $c$?
Spin comes from combining relativity with quantum mechanics.
I can't stress enough that this is the wrong idea about spin. Spin is an intrinsic property of the particles that have it, and has nothing to do with special relativity in any way.
You might be confused because the Dirac Equation incorporates quantum mechanics with special relatively and the spin of an electron can be derived from it, but a particle at rest (which is not really possible) has the same spin as an identical particle travelling at 99.9999% c.
In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.
Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus). The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.
In some ways, spin is like a vector quantity; it has a definite magnitude, and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number.
One way of looking at the Pauli exclusion principle is to think of it in terms of wavefunctions: half-integer spin particles must be described by antisymmetric wavefunctions, and particles of integer spin are required to have symmetric wavefunctions. The minus sign in the equation below implies the wavefunction must vanish identically if both states are "a" or "b", leading to the PEP, the law of nature that states that it is impossible for both electrons to occupy the same state in a bound system, that is, they cannot have the same 4 quantum numbers.
Does the Pauli Exclusion Principle still apply if c= ∞ or does it require a finite value for c?
The speed of light is not infinity, it is 299 792 458 m/s and it does not have any connection with the form of the wavefunction that describes the PEP.