Thanks, guys! That was really useful. I think that has helped me narrow it down to two possible culprits.
The bad news is that both were mods written by people in the paid work section here at vBulletin.org, and almost without exception, 100% of people hired there flake out and disappear. Which leaves me in something of a pickle...
What would you advise I do next? Try to hire someone to look at those two mods and see if they can figure out the issue?
Edit - confirmed; deactivating one of the two culprits fixes the issue, so at least is has been identified! The plugin itself is pretty short. Is it likely that the error is there in that short plugin, or will someone have to examine the entire mod?