We describe a novel search for MeV-to-GeV-mass dark matter, in which the dark matter scatters off electrons in a scintillating target. The excitation and subsequent de-excitation of the electron produces one or more photons, which could be detected with an array of cryogenic low-noise photodetectors, such as transition edge sensors (TES) or microwave kinetic inductance devices (MKID). Scintillators may have distinct advantages over other experiments searching for a low ionization signal from sub-GeV DM. First, the detection of one or a few photons may be technologically easier. Second, since no electric field is required to detect the photons, there may be far fewer dark counts mimicking a DM signal. We discuss various target choices, but focus on calculating the expected dark matter-electron scattering rates in three scintillating crystals, sodium iodide (NaI), cesium iodide (CsI), and gallium arsenide (GaAs). Among these, GaAs has the lowest band gap (1.52 eV) compared to NaI (5.9 eV) or CsI (6.4 eV), allowing it to probe dark matter masses possibly as low as ∼ 0.5 MeV, compared to ∼ 1.5 MeV with NaI or CsI. We compare these scattering rates with those expected in silicon (Si) and germanium (Ge). The proposed experimental concept presents an important complementary path to existing efforts, and its potential advantages may make it the most sensitive direct-detection probe of DM down to MeV masses.