In 2022, the James Webb Space Telescope (JWST) obtained 1-5um images of the center of the Orion Nebula Cluster (ONC). I have analyzed these data in an attempt to search for substellar members of the cluster. Using a pair of color-color diagrams, I have identified >200 brown dwarf candidates that lack spectral classifications. Some of the candidates could be protostars (either stellar or substellar) given their very red colors. Based on the age of the ONC and the photometry predicted by theoretical evolutionary models, the faintest candidates could have masses of 1-2M_J_. This sample of candidates may prove to be valuable for studying various aspects of young brown dwarfs, including their mass function and minimum mass. However, spectroscopy is needed to confirm the membership (via signatures of youth) and late spectral types of the candidates. Finally, I note that most of the "Jupiter-mass binary objects" that have been previously identified with these JWST images are absent from my sample of candidates because their colors are indicative of reddened background sources rather than young brown dwarfs, or their photometry is inadequate for assessing their nature because of very low signal-to-noise ratios and/or detections in only a few bands.