Whether a vaccine requirement is legal will depend on who is imposing it and how they have shaped it. Mandates that have been rolled out already appear to have been carefully designed with these factors in mind, though there are still some questions of how courts will handle more aggressive government approaches.
But, big picture, as the Delta variant and mediocre vaccination rates have changed the war on the coronavirus, policymakers have plenty of perfectly legal tools at their disposal for requiring the vaccine.
Private employers have the most discretion, from a legal perspective. According to [law professor Lindsay] Wiley, their main legal constraints are the Civil Rights Act of 1964, which prohibits religious discrimination, and laws that require accommodations for people with disabilities, like the Americans with Disabilities Act.
The US Equal Employment Opportunity Commission put out guidance laying out how employers can require vaccines while complying with those laws. The Justice Department, meanwhile, has announced that, in its interpretation of federal law, employers need not wait for the Food and Drug Administration to give the vaccines its full approval before requiring vaccines for workers.