So the traditional answer for these two groups (at least the ones in America) has always been "businesses have the right to fore any employee for any reason".
Now I get the sense that with the massive corporate abuse of power on free speech as well as this subject, opinions may be shifting.
But I am very interested in knowing what people think about this and why.
Assuming the shots were actually vaccines, and they were actually effective at stopping transmission, then yes. This would also assume any condition that prevents someone frome safely receiving the vaccine would be covered under ADA, and would be accommodated accordingly. Finally, any business mandating medical procedures for their employees would be liable for any injuries resulting from said medical proceedures.
All that said, the shots are not vaccines according to the 2019- definition, and the CDC says they are not effective at preventing transmission. Therefore, no company should be mandating the shots currently available.