If the requirement is imposed by an employer, then the basis for an exemption, if any, would be whatever the employer choose to allow, unless some applicable law required some particular exemption. Such mandates are not common. Indeed I do not recall hearing of any such absolute mandates. But an employer could choose to impose one, and I have heard (and read) discussions of hospitals, for example, requiring COVID-19 vaccinations to be taken by employees when available.
If an employer mandates a vaccination, it is up to the employer to decide when, and if, an exemption is warranted. It is up to the employee to decide whether to comply by accepting the vaccination or refuse at the risk of being fired.
I do not know of any law which would specifically require an employer to grant exemptions in particular circumstances. A law might exist, or be passed, requiring such exemptions.
If the employee had a particular medical condition which makes a vaccination si9gnificantly more risky for that employee than for an average person, and if that condition was considered to be a disability (not all or even most medical conditions are so considered) then the Americans with Disabilities Act (ADA) would require the employer to offer a "reasonable accommodation" to the employee, if one is available. Exactly what accommodation is "reasonable" is a fact-based determination, and would depend on the reason for the mandate, the cost and burden of the accommodation on the employer, and the degree if risk to the employee with and without the accommodation. In some cases no accommodation is found to be "reasonable", and in such cases the ADA does not mandate any accommodation at all.
The ADA generally specifies whet must be done when an employee has \ a disability, but dopes not specify exactly how a disability must be proved. A medical certificate is common, but njot invariable, iof an employer questions the existence o a disability.
The ADA would not require an exemption for a "Religious/Philosophical" objection. Soem state laws protect religious scruples in various particular employment situations. In some but not all states objections on such grounds for vaccinations otherwise required for school attendance are honored. The exact grounds accepted vary from state to state. A specific state would need to be listed for a more specific answer to be given, and I am not aware of any state that has such a requirement for exemption from a private employer's vaccine mandate.
Addition: It seems from comments that some employers, such as hospitals and the military, already require vaccinations for some diseases. This seems reasonable to me, and does not change my answer otehrwise.