They can do a criminal background check. Although now that is state dependent I believe.
The credit check is very iffy. Back when I was working for international cooperation it was disallowed in some states.
They can only have you meet certain health requirements if it directly impacts on your ability to do your job and even then they have to make "reasonable accommodations" if you need some sort of help to do it.
For example, they can require you to have good eyesight to drive one of their trucks but they can not say "good eyesight without glasses". The glasses are considered a reasonable accommodation.
Private companies can require that you not smoke at work or even away from work. With that, there will be companies that require employees to take a vaccine as a condition of hiring and employment.