In addition to what Wyorp said, a swollen vent may indicate that your chick has constipation problems. Watch this chick carefully. If it peeps rapidly as it tries to poop, that means there's a buildup of poop inside causing pain when it tries to poop, thus swelling the tissues around the vent.
By all means get as much liquid into the chick as possible. If the chick is eating, try mixing a little coconut oil or olive oil into a wet mash. Smear some of the coconut oil on the irritated vent to help soothe the inflammation.
If the chick has no appetite, I've had good luck using crumbled tofu instead of regular chick food. It's a pre-digested form of what's in chick food, and usually a sick chick will eat it when it won't touch its chick food. Also, get some poultry Nutri-drench and sprinkle it over the tofu or moistened chick food.
Be prepared, though, for this chick to go downhill. Many times when you have a chick with problems such as yours is having, it has genetic defects prevent normal processing and absorption of nutrients. Hopefully, this chick is a fighter and is going to pull through.
