Sub-GeV dark matter (DM) which interacts with electrons can excite electrons occupying molecular orbitals in a scattering event. In particular, aromatic compounds such as benzene or xylene have an electronic excitation energy of a few eV, making them sensitive to DM as light as a few MeV. These compounds are often used as solvents in organic scintillators, where the deexcitation process leads to a photon which propagates until it is absorbed and reemitted by a dilute fluor. The fluor photoemission is not absorbed by the bulk, but is instead detected by a photon detector such as a photomultiplier tube. We develop the formalism for DM-electron scattering in aromatic organic molecules, calculate the expected rate in p-xylene, and apply this calculation to an existing measurement of the single photo-electron emission rate in a low-background EJ-301 scintillator cell. Despite the fact that this measurement was performed in a shallow underground laboratory under minimal overburden, the DM-electron scattering limits extracted from these data are already approaching leading constraints in the 3-100 MeV DM mass range. We discuss possible next steps in the evolution of this direct detection technique, in which scalable organic scintillators are used in solid or liquid crystal phases and in conjunction with semiconductor photodetectors to improve sensitivity through directional signal information and potentially lower dark rates.