A company could tell everyone they hired for finance they now are being transferred to research and development and vice versa. Nothing against the law. But people in finance were not hired to do R&D and people in R&D were not hired to do financials. The company could then of course fire the above for not performing --but HR usually never takes this approach to layoffs. They may eliminate departments or redefine things and then you re-apply and so on with new roles and duties but it's all formal. Companies could require employees to re-apply, but you'd see quite some pushback especially from people who have been there a while and may interpret the move as a threat to their job security.
Yeah that's entirely permissible. My company can move me from engineering to toilet cleaning if they want. I would probably quit because I would think it's unreasonable, but I doubt I'd win any lawsuits. But I'd argue that this bank's decision is both permissible and reasonable.
In this situation, you have three groups of employees. Group A doesn't want the vaccine. Group B gets the vaccine but doesn't care whether Group A does or doesn't. Group C gets the vaccine and also doesn't want to work around people in Group A out of concern for their own safety. The company has to choose to accommodate either Group A or Group C. In this case, they chose the latter. Many companies will choose the former. But I don't see what's unfair about this? Group A might have started with the company with the expectation that vaccine status is private. But Group C might have started with the expectation that the company will take measures for their safety should situations change.
Note: I think it's debatable as to how many restrictions we should have in place because of Covid. But there's no denying that many people want to reduce their exposure to the virus and I don't think it's unreasonable for a private company to accommodate that.
Many U.S. states have labor law that cover this. And any company that I have seen do anything similar to this has lost money to both the state and the claimant.
Typically, at least for at-will states, the employer simply terminates you without stating any reason. Very simple for the company and the employee at least gets unemployment pay.