The MiniBooNE-DM Collaboration searched for vector-boson mediated production of dark matter using the Fermilab 8-GeV Booster proton beam in a dedicated run with 1.86×10^{20} protons delivered to a steel beam dump. The MiniBooNE detector, 490 m downstream, is sensitive to dark matter via elastic scattering with nucleons in the detector mineral oil. Analysis methods developed for previous MiniBooNE scattering results were employed, and several constraining data sets were simultaneously analyzed to minimize systematic errors from neutrino flux and interaction rates. No excess of events over background was observed, leading to a 90% confidence limit on the dark matter cross section parameter, Y=ε^{2}α_{D}(m_{χ}/m_{V})^{4}≲10^{-8}, for α_{D}=0.5 and for dark matter masses of 0.01<m_{χ}<0.3 GeV in a vector portal model of dark matter. This is the best limit from a dedicated proton beam dump search in this mass and coupling range and extends below the mass range of direct dark matter searches. These results demonstrate a novel and powerful approach to dark matter searches with beam dump experiments.