We present the first catalogs of the highest-energy (above 56 TeV and 100 TeV) gamma-ray sources seen by the High Altitude Water Cherenkov (HAWC) Observatory. The wide field-of-view of HAWC naturally lends itself to unbiased all-sky surveys and newly developed event-by-event gamma-ray energy reconstruction algorithms have allowed unprecedented energy resolution. The sources presented here are the highest-energy sources ever detected. All are coincident with known lower-energy gamma-ray sources within our Galaxy. These objects may have implications for the sources of Galactic cosmic rays; since Galactic CRs have been observed up to PeV energies, sources accelerating particles to these energies must exist. These sources, called "PeVatrons'', would have corresponding hard gamma-ray spectra that extend to high energies without any spectral break or cutoff. We will present measurements of the spectra of these highest-energy gamma-ray sources and discuss if any of them can be identified as PeVatron candidates.