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).
A gentler, a less jargon filled explanation would be as follows: Technically there is nothing prohibiting a massless particle from carrying charge. The intuitive notion of a massless particle being affected by radiation continues to hold in the standard model. On the other hand, spin is subtler in that it requires you to think about what exactly is spin. One way to think about spin is to apply a magnetic field to the particle in its rest frame and ask how many possible configurations the particle has. However, for massless particles, relativity prohibits you from catching upto the particle and thinking of a rest frame for the particle. For massless particles the notion of spin is replaced by helicity.