What are the e risks of contracting Covid at a vaccination site?

The risk is minimally higher than not being in the presence of others. You are, however, at far greater risk of catching COVID-19 in a supermarket.

At a vaccination centre people are given appointments so they do not arrive en masse. People are requested to wear a face mask or other face covering (unless medically exempt); they are asked to use hand sanitiser; they are asked to keep 2 m apart. There are stewards at the vaccination centre to coordinate these safety measures.

What packed crowds??

The place I went to had appointments scheduled every ten minutes so not many people were there at a time.

Everyone arriving had their temperature checked before being let in, and were required to use hand sanitiser (and be wearing a mask).

There was a waiting room after booking in, with well spaced chairs and someone cleaning each chair as someone got up.

The room with the doctors/nurses had positions several yards apart for the injections, and again everything was sanitised as soon as anyone left.

I've had both shots now, a few weeks apart - and the setup was the same for both.

So no - the chances of cross-infection were zero, from what I saw.

Almost zero since those locations have strict protocols to keep everyone safe.

They keep you spaced out. You'll be fine