Is spin necessary for electromagnetism?

Macroscopically, in classical mechanics spin is the rotation of a body around an axis passing through it, and can be seen as the angular momentum that the body has. Conservation of angular momentum is intrinsic in the classical theory .

When measurements started to look at the microcosm of atoms and particles, the conservations rules of energy, momentum were found to hold in the interactions, BUT unless an intrinsic angular momentum was given to particles, conservation of angular momentum would not hold. So by experimental observations a fixed angular momentum, called spin, was assigned to all elementary particles so that the observations would fit the quantum theory that was developing. This has been validated over and over again with all measurements.

You ask:

how big of an role does spin play in the interactions which happen through the electromagnetic field?

The big role of keeping conservation of angular momentum in interactions .

Are there some interactions which spin-less particles cannot have, but those with spin can?

The fundamental interactions of elementary particles are four. Spin limits the possible interactions because of conservation of angular momentum, so there are differences between spin-non-zero-particle interactions and spin-zero-particles, as with all conserved quantities, like charge, baryon number etc.

The charged pions, for example, $\pi^+$ and $\pi^-$, have zero spin but interact with a magnetic field, as can be seen from the curved tracks they leave in a bubble chamber with a magnetic field. So, to answer your question, spin is not theoretically necessary for electromagnetism. You could have a perfectly good and complete EM theory with spinless particles. But the real world doesn't work like that.

Classical electrodynamics is formulated in terms of macroscopic (i.e. averaged over many atoms/particles) fields and sources (currents and charges). It is fully condensed in Maxwell equations and supporting material equations (describing how the sources respond to the fields). As such it does not need spin, simply because it doesn't care about the origin of the magnetic moments involved. Indeed, interpreting spin in macroscopic terms, as a current due to the particle rotation, is known to be incorrect: quantitatively for the charged particles, and qualitatively for the neutral ones (such as neutron).

From the quantum electrodynamics point of view, the spin is a distinction between the cariers of interaction (bosons, which have integer spin, such as photons) and the fermions that couple to the carriers of interaction via their charge and spin. It is not quite clear how one can throw away the spin in this picture without destroying it completely.

Finally, spin affects the interactions inexplicitly via the particle statistics, i.e. via the exclusion principle.