Why is my kitten drinking so much water after a recent vet visit?

That certainly does sound like a big day for JJ! Vaccines, deworming and flea medication, and leg injuries don't cause sudden increases in water intake. Now, if JJ was kept at your vet in a cage without water for several hours for a drop off appointment, then it would be reasonable to expect him to drink a lot of water at home. If he wasn't dropped off away from water, I'd be more inclined to suspect another issue, such as a urinary tract infection. Diabetes and kidney disease can also cause pronounced increases in water consumption, but they would be highly unusual in a kitten as young as JJ. If JJ continues to drink excessive amounts of water tomorrow, then I would certainly take him back to the vet for evaluation. Your vet will likely want to at a minimum check a urine sample for evidence of infection. I hope that everything goes well!