We present observations of the two closest globular clusters, NGC 6121 and NGC 6397, taken with the NIRISS detector of JWST. The combination of our new JWST data with archival Hubble Space Telescope (HST) images allows us to compute proper motions, disentangle cluster members from field objects, and probe the main sequence (MS) of the clusters down to <0.1 M⊙ as well as the brighter part of the white-dwarf sequence. We show that theoretical isochrones fall short in modeling the low-mass MS and discuss possible explanations for the observed discrepancies. Our analysis suggests that the lowest-mass members of both clusters are significantly more metal-rich and oxygen-poor than their higher-mass counterparts. It is unclear whether the difference is caused by a genuine mass-dependent chemical heterogeneity, low-temperature atmospheric processes altering the observed abundances, or systematic shortcomings in the models. We computed the present-day local luminosity and mass functions of the two clusters; our data reveal a strong flattening of the mass function indicative of a significant preferential loss of low-mass stars in agreement with previous dynamical models for these two clusters. We have made our NIRISS astro-photometric catalogs and stacked images publicly available to the community.