He was admitted to an emergency 24hr vet clinic. They couldn't find a direct source, but they decided that he definitely has an upset stomach. They claimed it was aggravated by several things, potentially including the fact he was suddenly switched off of kitten food --> adult food. We were told we'd have to feed him human baby food for a few days.
They've got him on several medications, just as a precaution for worms & other things that could make him sicker. But it seems like he should be okay soonish.