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We report the first results of a search for leptophobic dark matter (DM) from the Coherent CAPTAIN-Mills (CCM) liquid argon (LAr) detector. An engineering run with 120 photomultiplier tubes (PMTs) and $17.9 \times 10^{20}$ protons-on-target (POT) was performed in Fall 2019 to study the characteristics of the CCM detector. The operation of this 10-ton detector was strictly light-based with a threshold of 50 keV and used coherent elastic scattering off argon nuclei to detect DM. Despite only 1.5 months of accumulated luminosity, contaminated LAr, and non-optimized shielding, CCM's first engineering run already achieved sensitivity to previously unexplored parameter space of light dark matter (LDM) models with a baryonic vector portal. With an expected background of 115,005 events, we observe 115,005+16.5 events which is compatible with background expectations. For a benchmark mediator-to-dark matter mass ratio of $m_{_{V_B}}/m_{\chi}=2.1$, DM masses within the range $9\,\text{MeV} \lesssim m_\chi \lesssim 50\,\text{MeV}$ have been excluded at 90% C.L. in the leptophobic model after applying the Feldman-Cousins test statistic. CCM's upgraded run with 200 PMTs, filtered LAr, improved shielding, and ten times more POT will be able to exclude the remaining thermal relic density parameter space of this model, as well as probe new parameter space of other leptophobic DM models.