We present the first results from the Revealing Low-Luminosity Active Galactic Nuclei (ReveaLLAGN) survey, a JWST survey of seven nearby LLAGNs. We focus on two observations with the Mid-Infrared Instrument (MIRI)'s Medium-Resolution Spectrometer of the nuclei of NGC 1052 and Sombrero (NGC 4594/M104). We also compare these data to public JWST data of higher-luminosity AGNs, NGC 7319 and NGC 7469. JWST clearly separates the AGN spectrum from the galaxy light even in Sombrero, the faintest target in our survey; the AGN components have very red spectra. We find that the emission-line widths in both NGC 1052 and Sombrero increase with increasing ionization potential, with FWHM > 1000 km s‑1 for lines with ionization potential ≳ 50 eV. These lines are also significantly blueshifted in both LLAGNs. The high-ionization-potential lines in NGC 7319 show neither broad widths nor significant blueshifts. Many of the lower-ionization-potential emission lines in Sombrero show significant blue wings extending >1000 km s‑1. These features and the emission-line maps in both galaxies are consistent with outflows along the jet direction. Sombrero has the lowest-luminosity high-ionization-potential lines ([Ne V] and [O IV]) ever measured in the mid-infrared, but the relative strengths of these lines are consistent with higher-luminosity AGNs. On the other hand, the [Ne V] emission is much weaker relative to the [Ne III] and [Ne II] lines of higher-luminosity AGNs. These initial results show the great promise that JWST holds for identifying and studying the physical nature of LLAGNs.