Lots of jobs have medical requirements, including requirements to take any medications prescribed to you, including, at the simplest having to meet certain eyesight requirements to drive, this includes having to wear glasses/contact lens to meet these standards.
Lorry drives also have to meet stringent health requirements. If you go offshore you need to produce a certificate of dental fitness, showing that you have been to a dentist and had any work required done. Saying you have an objection or phobia to going to the dentist and having treatment, won't cut it. Either you get the certificateand have the treatment or you do not go offshore.
Requiring that you have the COVID and I would include flu vaccines if you want to work in a care home seems to me entirely reasonable. If you do not want them, then work elsewhere.
Tell me biscuitmuncher If you were frail and in care or someone very dear to you was old and frail and in care, would you be happy to know that several of the carers looking after you had not been vaccinated against COVID and could give you the disease or your loved one a disease that has a high chance of proving fatal?
Please do not duck this question, the way most vaccine denier/doubters do. Prove you are not another of the gutless womders.