To challenge such firing successfully, it will be up to the employee to find a specific law prohibiting them from being fired for refusing to take the vaccine. There isn't any federal law that says that clearly and unambiguously, so they're going to have to find other statutes and argue that they apply in their case.
Some employees may win, but it isn't going to be a slam dunk.
Can a company, lawfully, mandate all employee's get an influenza vaccine as a condition of their employment?
Can a company, lawfully, mandate every employee have their blood pressure recorded by a health care provider, and report their results back to HR?
Can a company, lawfully, mandate all male employee's get a prostate examination, or all female employee's get a mammogram, as a condition of their employment?
Can a company, lawfully, mandate all employee's receive a herpes zoster vaccination as a condition of their employment?
Can a company, lawfully, mandate all employee's take an aspirin once a day as a condition of their employment?