This is basic law: Stuff is allowed unless forbidden. Because of that, employers generally have the right to require things of employees, and the exceptions are what's specified in law.

So if your employer wants to require of the people in your team that they know C++, speak Swahili or carry photo ID, that's allowed because not forbidden. But the law does have provisions for religious discrimination (in most countries, perhaps all) so requiring your team to be Pastafarians is out (if that's what the law says, the law might make an exception for the Pastafarian Church).

Requiring vaccines might also be forbidden by your employment contract. But is it? I doubt it.

Your contract also could say "the employer cannot permit you to take paid sick leave if you're ill on the day after a vaccine" or "the employer cannot permit you to get a vaccine on company time" but both sound far-fetched to me.