They can't force you. But they can say you can't work there unless you do.
Everyone is forgetting the preliminary problem: they are asserting the right to ASK you a private health question. I don't answer those, ever, ever, ever. People need to mind their own damn business around me from now on. So forget about the shot: I'd quit if they insist on an answer to an inappropriate question. I think they'd have to fire me, actually. Might get something out of that if there's a class-action suit.
Also, they wouldn't just believe someone: you'd have to show your "vaccine passport." That is not happening with me.
Easy to find a job these days! Everyone is begging for workers. Pretty crazy to throw good workers away on such a civil liberties issue, especially considering how masks don't work, the vaccine failed, etc. And people will remember obnoxious employers that don't allow privacy, restaurants that are into power trips against their customers, stores with all sorts of horrible rules, stand here, don't stand there -- I don't want anything to do with totalitarian control freaks.