We study the sensitivity of the Fermilab Short-Baseline Neutrino (SBN) experiments, Micro-BooNE, ICARUS, and SBND, to MeV-to GeV-scale inelastic dark matter interacting through a dark photon mediator. These models provide interesting scenarios of light thermal dark matter, which, while challenging to probe with direct and indirect detection experiments, are amenable to accelerator-based searches. We consider production of the dark sector states with both the Fermilab Booster 8 GeV and NuMI 120 GeV proton beams and study the signatures of scattering and decay of the heavy excited dark state in the SBN detectors. These distinct signatures probe complementary regions of parameter space. All three experiments will be able to cover new ground, with an excellent near-term opportunity to search for cosmologically motivated targets explaining the observed dark matter abundance.