Spallation neutron sources provide intense neutrino fluxes from pion and muon decay at rest, with energies in the few tens of MeV range. Experiments such as COHERENT exploit these fluxes to detect neutrinos via coherent elastic neutrino–nucleus scattering (CE$\nu$NS). However, these facilities also offer a unique opportunity to produce and probe light, secluded, or weakly-coupled particles. In this work, we investigate the sensitivity of the Spallation Neutron Source at the Oak Ridge National Laboratory to heavy neutral leptons (HNLs) in the MeV-GeV mass range, as a case study. We consider HNL production in pion and muon decays at rest, followed by their decay into visible Standard Model particles within the detector volume. We analyze a range of current and proposed COHERENT detectors and evaluate their sensitivity to HNL mixing with muon and electron neutrinos, as well as to scenarios with mixed mixing. We find that existing detectors can set meaningful constraints, particularly for muon-flavor mixing, while future ton-scale realizations can probe previously unexplored regions of the parameter space. We further discuss prospects at other relevant spallation source facilities and comment on the complementarity of their projected sensitivities. Our results demonstrate that CE$\nu$NS experiments at spallation neutron sources provide a powerful, complementary avenue for new physics searches beyond their primary role as neutrino detectors.