Experiments aiming to detect coherent neutrino-nucleus scattering present opportunities to probe new light weakly-coupled states, such as sub-GeV mass dark matter, in several extensions of the Standard Model. These states can be produced along with neutrinos in the collisions of protons with the target, and their production rate can be enhanced if there exists a light mediator produced on-shell. We consider the sensitivity reach of the COHERENT experiment at SNS, which recently observed coherent neutrino-nucleus scattering, to light dark matter interacting with the Standard Model via a light vector mediator coupled to the electromagnetic current.