With the classical theory of electrodynamics, something like the hydrogen atom is not stable. Non-relativistic quantum mechanics is able to give a stable description of electrons in a hydrogen atom, and we can solve for the hydrogen energy levels. Furthermore, basically all the intro chemistry descriptions of atomic orbitals, such as S vs P vs D orbitals and the basics of bonding, come from non-relativistic quantum mechanics and extrapolating from those single electron hydrogen like wavefunctions.
There are many computational chemistry models that don't include relativity explicitly. One caveat that I'd mention is that they do include it implicitly as soon as there are multiple electrons. For example the Hatree-Fock method describes molecular structures by describing molecular orbitals in some basis (a linear combination of atomic oribitals about each atom, or gaussian approximations of atomic orbitals, etc). The wavefunction is then written as a Slater determinant of these molecular orbitals. With only a single Slater determinant, the majority of the electron-electron correlation is ignored. Basically, the only correlation you get is that, due to the structure of a determinant, it provides that the probability that two electrons of the same spin are found at the same location is zero. These somewhat crude models have been quite successful in the initial exploring of computational quantum chemistry.
In other words, the quantum chemistry models can be quite successful with just non-relativistic quantum mechanics and without including relativistic terms or corrections in the Hamiltonian (there are not even magnetic interactions in Hartree-Fock, the momentum operation is non-relativistic, and so on). However, relativity is implicitly snuck in at the fundamental level by using the Slater determinants. This is what enforces the Pauli exclusion principle. It is essentially put in by hand.
Without relativity, you would not be able to derive the spin statistics theorem. This is incredibly important to the structure of molecules (otherwise all the electrons would essentially just pile up in the same lowest energy orbital).
That being said, I see no reason you couldn't just postulate spin and the Pauli exclusion principle, and let non-relativistic quantum mechanics take you the rest of the way to quantum chemistry.
Alternatively, if you are worried about quantizing the electromagnetic field and getting into quantum field theory, I see no reason you could not (at least in principle) just obtain a well defined theory by taking the $c \to \infty$ limit of our current theories. In the $c \to \infty$ limit, special relativity reduces to Galilean relativity of Newtonian mechanics. This limit is a bit counter-intuitive in that some everyday things may change, so I cannot speak confidently on the results. For instance I think the effect of the magnetic field drops to zero as the speed of light is taken to infinity, and I'm not sure if radio communication could work anymore. If you are just fantasy "WorldBuilding" and going full Newtonian, you could try to use the old idea of a medium for electromagnetism and have the speed of light still be finite (the same way sound propagating in a material is much less than c), even if the relativity parameter "c" goes to infinity.