In the standard model, every massless particle transforms under representation of $U(1)_{EM}$ group. Therefore, every particle has an electric charge (some of them transform in the trivial representation and therefore have zero electric charge). On the other hand, spin is not well defined for a massless particle, since it's a quantity natural to the rest frame of the particle. Instead massless particles carry helicities, which are analogous to spin, although the word spin continues to be used colloquially. This is because massive particles transform in representations of their little group $SO(3)$ a.k.a spin, but massless particles transform in representations of $SO(2)$ a.k.a helicity.
So to answer your question: massless particles carry can charge but not spin (in the strictest sense of the word).