Earlier quoted context omitted.
There's long established precedent that schools and colleges are allowed to require proof of vaccination. The idea isn't remotely new, and is nothing so odious as critics claim
I am not a lawyer, nor a public health legal expert but I think there is a significant legal difference between COVID and ordinary vaccine requirements because the COVID vaccines are only authorized for emergency use. I think schools and colleges requiring vaccination are also offering accommodations (distance learning, etc.) to avoid legal fights, while if the vaccines were fully authorized they would not.
This is only a technical, legal distinction. They have been administered rapidly to a significant fraction of the population; their specific legal status is mostly irrelevant at this point.
They are GA in practice, the release has been cut and people are running it. Paper pushers slapping an ass-covering "beta" on the splash screen are behind the times.