An employer requiring vaccinations prior to employment isn't unheard of in the US. It has been going on for several decades for workers in the education and healthcare industry as some of them have had to get vaccinated for chickenpox, measle, mumps, rubella, the flu, and other viruses as a requirement for employment at their respective organizations.

https://www.google.com/amp/s/amp.usatoday.com/amp/3346210001

Also, what would be the next part after this? We would want to first evaluate that before allowing them access to it.

As for remote workers, I think that is a legitimate point and something companies should consider.