So, I decided to try and confirm the diagnosis before hunting the stuff down. I held the chick and opened its mouth while my wife looked down its throat to see if she could spot the gapeworms. Gapeworms are hooked together in a "V", in permanent conjugal bliss... at least, I guess it's bliss.

Here's a romantic picture of a bunch cuddling up in a chicken's trachea
Anyway, my wife couldn't find anything in our chick's throat. So now I'm back to wondering what the heck the sneezing is. Maybe something I need to treat, maybe not. I'll watch and see if any of the other birds start sneezing too.