I *think* some vaccinations are already mandatory for school. I'm not sure on that. I know a friend of mine tried to avoid some vaccinations for her daughter and was basically unable to do so.
As far as a COVID 19 vaccination is concerned, it could be mandatory for school and college, but I doubt if it could go beyond that.
The elephant in the room is that very few people are going to take the vaccination anyway. Only 37% of adults take the flu shot AND the percentage is going down each year recently instead of up. I doubt if more than 20% of adults will get the COVID 19 vaccination, at least for the first year that it's out. You have the "I took the flu shot and got the flu so never again" crowd, the anti-vaxxer crowd, the "I never get the flu so I'm not getting the shot" crowd, and those who just can't be bothered with getting a shot or are fatalistic "if I get it I get it". To those groups, you can add the new "it's the Trump vaccine" crowd and the "it's been rushed to market without proper long term testing" crowd.
I doubt if the vaccination rate will be more than 1 in 5 adults, excluding those who must take it involuntarily.
Heck, the vaccines aren't even being tested against the real virus. They're just being tested to show the creation of antibodies. We don't know how long antibodies will last. It could only be a matter of weeks. We also don't know with certainty if they will prevent COVID in the average (read that as sedentary, overweight, and deficient in many vitamins) person.