We use data from the COHERENT CsI[Na] scintillation detector to constrain sub-GeV leptophobic dark matter models. This detector was built to observe low-energy nuclear recoils from coherent elastic neutrino-nucleus scattering. These capabilities enable searches for dark matter particles produced at the Spallation Neutron Source mediated by a vector portal particle with masses between 2 and 400 MeV/c2. No evidence for dark matter is observed and a limit on the mediator coupling to quarks is placed. This constraint improves upon previous results by two orders of magnitude. This newly explored parameter space probes the region where the dark matter relic abundance is explained by leptophobic dark matter when the mediator mass is roughly twice the dark matter mass. COHERENT sets the best constraint on leptophobic dark matter at these masses.