We analyze Chandra X-ray spectra of the M0 V+M0 V binary GJ 338. As quantified by X-ray surface flux, these are the most inactive M dwarfs ever observed with X-ray grating spectroscopy. We focus on measuring coronal abundances, in particular searching for evidence of abundance anomalies related to first ionization potential (FIP). In the solar corona and wind, low-FIP elements are overabundant, which is the so-called FIP effect. For other stars, particularly very active ones, an "inverse FIP effect" is often observed, with low-FIP elements being underabundant. For both members of the GJ 338 binary, we find evidence for a modest inverse FIP effect, consistent with expectations from a previously reported correlation between spectral type and FIP bias. This amounts to strong evidence that all M dwarfs should exhibit the inverse FIP effect phenomenon, not just the active ones. We take the first step toward modeling the inverse FIP phenomenon in M dwarfs, building on past work that has demonstrated that MHD waves coursing through coronal loops can lead to a ponderomotive force that fractionates elements in a manner consistent with the FIP effect. We demonstrate that in certain circumstances this model can also lead to an inverse FIP effect, pointing the way to more detailed modeling of M dwarf coronal abundances in the future.