I don't believe there is a massless charged particle, but would it even be possible for a massless particle to be charged as the energy of the charge itself would be a kind of non-momentum energy and therefore all of it's energy would not be in the form of momentum which would imply sub-light velocity and mass.
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1$\begingroup$ You might want to specify whether by "charge" here you mean electromagnetically charged or a more general charge, since e.g. gluons are massless and color-charged. $\endgroup$– ACuriousMind ♦Commented Mar 24, 2020 at 16:21
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It is true that there are no massless charged particles in the standard model (the case for Yang-Mills is not settled yet, but there are strong indications that the lightest particle is indeed massive). However, if one asks whether it is possible to have a massless charged particle, we are really free to answer yes if it is possible to write down a consistent quantum field theory that describes such a thing. This is indeed so, supersymmetric gauge theories will admit massless charged states, as will conformal field theories. The various "non-momentum contributions" to the energy balance out in such theories so that the lightest excited state is massless.
