Although The Joint Commission (TJC) no longer requires you to collect proof of vaccination, they do require you to track data around why people are NOT getting vaccinated. Additionally, you should be reviewing this data before Flu season each year. Your Annual Flu program should plan to provide education to staff, using the declination reasons to empower and encourage them to get vaccinated.
Ultimately, you need to see what your State regulations say about Influenza and COVID19 vaccinations and Tuberculosis screens. States such as New Jersey have clear requirements that very specific requirements around Tuberculosis (TB), Measles Mumps Rubella (MMR), and Physical Examinations.