The Mu2E and COMET $\mu \rightarrow e$ collaborations plan to advance branching ratio sensitivities by four orders of magnitude, further constraining new sources of charged lepton flavor violation (CLFV). Here we formulate a non-relativistic nucleon-level effective theory (ET) for this process, in order to clarify what can and cannot be learned about CLFV operator coefficients from elastic $\mu \rightarrow e$ conversion. Utilizing state-of-the-art shell model wave functions, we derive bounds on operator coefficients from existing $\mu \rightarrow e$ conversion and $\mu \rightarrow e \gamma$ results, and estimate the improvement in these bounds that will be possible if Mu2E, COMET, and MEG II reach their design goals. In the conversion process, we employ a treatment of the lepton Coulomb physics that is very accurate, yet yields transparent results and preserves connections to standard-model processes like $\beta$ decay and $\mu$ capture. The formulation provides a bridge between the nuclear physics needed in form factor evaluations and the particle physics needed to relate low-energy constraints from $\mu \rightarrow e$ conversion to UV sources of CLFV.