how likely are you to catch covid at a vaccination site?

The conditions in the vaccination center that I observed when I got my shot were pretty much paranoiac: first, the mask one wore coming it was replaced by one of theirs, they wiped clean the chair you just sat on with disinfectant, they didn't even touch the id card unless through a wet wipe; with the exception of the person giving the injection, nobody got even near me.

Overall, I would say if one catches it there, they would have had to bring it with them in the first place.

Less likely than you are to be hit by lightning on the way to the vaccination site. 

100's packed into a building at the same time, 1000's passing through during the day. Don't touch anything, keep your masks(two) on, and pray.

health care workers are scrupulous at safety. The odds are about zero. They will be in masks and gowns for sure. you are spaced apart.

Not likely. You’re not getting a shot up your nose. It’s on your arm. You’ll be wearing your face mask. People are kept 6 week apart at the monitoring room after the shot. That’s if you live in a state where masks are still required. If you live in Texas, Mississippi, and Florida, you’re more likely since there’s no mask to protect you.