Depends on the job. Healthcare workers, particularly those who work in hospitals, are required to be up to date on all their vaccinations as a condition of employment. Teachers and school staff as well in most places. There are some other examples, but I think those are probably the largest.
A lot of it comes back to limiting liability. If they don’t require it and someone gets seriously ill or dies, that’s a massive civil liability sitting out there for them.
I believe you have explained the question quite well.
In short, yes an employer can require the their employees be vaccinated.