Due to the variety of regional cultures, if the decision is left to the states (which it must be in this system), there will be some that would and some that would not require the Covid vaccine, eventually. I suspect relatively few would, actually. And there'll be serious howling and court challenges if they did. Back in the 1800s, they did make vaccinations for smallpox mandatory in some parts of the USA. Polio vaccine was widely mandated for all children quickly after it was developed - if they didn't have it, they were not allowed to attend school. (I think that one is still in effect.) I know that in at least two states in which I have lived, if children are not immunized against measles, pertussis, and, I think, diptheria the same applies - no shots, no school. There were exceptions for kids with medical reasons. In one place I lived for a while there used to be allowances for religious beliefs, but I think they've recently eliminated that. Parents can circumvent the requirements quite often by home-schooling or moving to a different state that allows exceptions on religious or personal grounds. Of course, if the kids eventually want to go to University, they're screwed in that case - I don't know of any that do not require full immunization of students for the major historical diseases like polio/measles, etc.
I don't know of a lot of mandates for vaccinating adults, historically, though. There will never be a federal vaccine order impacting civilians (that can be enforced) - the governmental system doesn't work that way here. It'll be up to the individual states to enact. (Interesting discussion -
https://www.cdc.gov/vaccines/imz-managers/guides-pubs/downloads/vacc_mandates_chptr13.pdf )
Employers can require employees be vaccinated in the USA. (It's already been through the courts). My former employer is doing just that - you can't set foot on the property without being vaccinated against Covid, as an employee. All the students... there is no requirement that I know of, yet, that the students be vaccinated. So I guess they just don't want to have to hire and train new staff if the existing staff had to take time off work or be let go if they used up their "time off" allowance. Knowing my ex-employer, the $$$ is the bottom line. They would have to hire temps to fill in, and the jobs tend to be a bit esoteric so you either have to pay premium to get someone who is actually competent, or (as is usually the case) you get a minimum-wager without prior training and just kind of limp by. There's a critical mass of those kind of employees they have to avoid if they want to keep the place a going concern, though. Hence, the staff vaccination requirement. (Or, that's my guess on it, anyway.) There are always plenty of students waving money, wanting to get into the school, so they are of less concern.
Major difference in the UK and USA in the above - the UK has a centralized government that has pretty much unlimited power to decree and enforce (devolution complicates, but only marginally). The USA's federal government has that power in some instances, but not nearly as broadly. The individual states here have much more control over the minutiae of life.
But as to my original post: My issue is with the military. They really do need to get those guys vaccinated and if it takes a formal order, they need to do that ASAP. Otherwise, in time of need (and one never knows when that could happen) we could theoretically find that we have half the military strength we need. I wouldn't want to think about it ripping through, for example, the drone control room in Kansas (or wherever it is) and the second or third-string trainees having to be brought in to manage all that in time of war. Or flying a warbird armed with God only knows what. Etc.
[The Ukraine issue is looking a little iffy again, lately.]