Now that the FDA has given final approval to the Pfizer vaccine, many employers have started requiring employees to get the covid vaccine. Can employers require employees to get a coronavirus vaccine? It is very likely that they can. We start with the general OSHA requirement of all employers to maintain a safe workplace. Every employer must take steps to prevent known dangers in the workplace. Employers do need to make allowances for religious beliefs and medical requirements. But, otherwise, there is no law or regulation preventing employers from requiring the vaccine.
One state, Montana, has passed a statute preventing employers from requiring the vaccine. But, in every other state, there is no obstacle to employers requiring a vaccine. And, the rule in some 40 states is at-will employment. At-will employment means an employer can generally do as it wishes. If an employee objects, then s/he is free to quit.
See ABA Bar association article here.