Quantum mechanics is much more general than the Schrodinger equation, which first introduced the wave function concepts . The postulates of quantum mechanics are much more general, and different wave equations are used for different situations.
For relativistic wave functions one uses Klein Gordon for bosons, Dirac for fermions and a quantized version of Maxwell's equation for photons. The last is not very often stressed because, as you note, photons are relativistic and quantum field theory is used directly, without elaborating how the ground state function of the photon field is obtained. Here is a presentations on how the photon wave function is. Even at low energy experiments, as the double slit, the photon is described by its appropriate wave function .
Quantum field theory is a mathematical method of dealing with many body interactions in the quantum mechanical regime, but it is based on the postulates of quantum mechanics. It defines particle fields in all of spacetime and creation and annihilation operators for all these fields which act on the ground state wave function of the field, taken from the solution of the free equations mentioned above . The photon is treated in the same way as all the other particle/fields in the elementary particle table .
The interaction crossections written in the QFT formalism allow for the calculation of observables in experiments in the relativistic regime.