I am not a lawyer so don’t rely on my legal reasoning here, but I am not sure specifically in this case that a company can require you to get this vaccine even after it is approved. Even with at will employment there must be limits on what an employer can fire you for. For instance could he require you to stand on your head every evening for ten minutes? Could he fire you for voting the “wrong way “ or not voting at all. I’d like to see the case law.
If we grant that an employer cannot fire you for things that do not affect your performance or the firms reputation or the companies needs i.e. your job is redundant then we have to see what grounds the employer is using.
The obvious answer is the unvaxxed employee poses a threat to customers and fellow employees. That however raises the question of not only does he pose a threat but does he pose a greater threat than a fully vaccinated employee and that might be harder to prove than at first appears. The Delta variant looks to be avoiding this vaccine at a very high rate. If the vaxxed spread covid too can you treat the unvaxxed employee differently and fire him? I don’t know.
I am trying to focus on the big picture—the original poster needs (at a minimum) a fallback position (a new job, career) if he does not want to get the jab under duress.
That type of change is very stressful, but there are tricks to help make it happen.