Of course they do. That’s fine.
Make it a condition of employment, or not.
But don’t make out people should be having the vaccination because they should ‘care’ and if they don’t they are in the wrong job.
Care workers are both badly paid and badly treated. It is morally reprehensible to pay them peanuts, demand they deal with shit, death and disease with a smile on their face, and then crossly say that they should ‘care’.