Dark matter is typically assumed not to couple to the photon at tree level. While annihilation to photons through quark loops is often considered in indirect detection searches, such loop-level effects are usually neglected in direct detection, as they are typically subdominant to tree-level dark-matter-nucleus scattering. However, when dark matter is lighter than around 100 MeV, it carries so little momentum that it is difficult to detect with nuclear recoils at all. We show that loops of low-energy hadronic states can generate an effective dark-matter-photon coupling, and thus lead to scattering with electrons even in the absence of tree-level dark-matter-electron scattering. For light mediators, this leads to an effective fractional electric charge that may be very strongly constrained by astrophysical observations. Current and upcoming searches for dark-matter-electron scattering can thus set limits on dark-matter-proton interactions down to 1 MeV and below.