High-energy beams incident on a fixed target may scatter against atomic electrons. To a first approximation, one can treat these electrons as free and at rest. For precision experiments, however, it is important to be able to estimate the size of, and when necessary calculate, subleading corrections. We discuss atomic binding corrections to relativistic lepton-electron scattering. We analyze hydrogen in detail, before generalizing our analysis to multi-electron atoms. Using the virial theorem, and many-body sum rules, we find that the corrections can be reduced to measured binding energies, and the expectation value of a single one-body operator. We comment on the phenomenological impact for neutrino flux normalization and an extraction of hadronic vacuum polarization from elastic muon electron scattering at MUonE. Published by the American Physical Society 2024