If you're in a business where that's likely, guess what, mandatory HIV tests.
But often prohibited during work, and when not generally not a job requirement, and when it is a requirement of a (legal) job, there's also a whole lot of intrusive requirements that would generally be considered peivacy violations in other work.
Breathing the same air as coworkers, though, is a practical requirement of in-office jobs, not something that might occasionally occur as an extracurricular activity during working hours despite being generally prohibited by policy or, if not, outside of requirements.
Do see now how requiring a vaccine because it would mitigate disease transmission risk in real world scenario that's a requirement of in person work is reasonable, vs requiring it because two people might sneak off to the head to get the leg over?