Lepton number violating (LNV) $A(e^{-},e^{+})X$ double charge exchange (LDCE) reactions on nuclei at accelerator facilities with multi-GeV beams are proposed as a probe for physics beyond the Standard Model (BSM). A second order formalism is presented including LNV dynamics by the left-right symmetric model (LRSM). For practical studies a phenomenological model is used to estimate LDCE cross sections numerically. Sizable cross sections are predicted for multi-GeV beam energies. LDCE reactions proceed preferentially by the energy-momentum dependent left-right mixing terms. While Majorana mass terms are negligible for light neutrinos, they may become sizable for heavy neutral leptons at higher beam energies. In the 10~GeV region inclusive total LDCE cross sections of about $100\times |{\Gamma }_{BSM}{|}^2$~fb in units of the BSM vertices are predicted, increasing strongly with energy and target mass. LDCE experiments seem to be feasible with existing equipment under full laboratory control and free of the constraints imposed on decay or capture experiments of nuclear and hadronic systems.