The Constitutional anti-discrimination protection is the 14th amendment, and it applies to federal state and local governments, but not to a private employer - it's no help if an employer wants to discriminate in their hiring and firing practices.

Anti-discrimination laws do apply to everyone, not just the government. But there's no law banning discrimination on the basis of vaccination status yet, so that's no help either.

Edit: Aside from getting your lawmakers to pass a law prohibitig employment discrimination on the basis of vaccination status, there is as another possibility: a good faith religious objection to vaccination might work, because employment discrimination on the basis of religion is not allowed.