There's plenty of precedent to requiring vaccinations, especially in schools. And the US government has already stated that employers requiring a vaccination in this case is generally OK.
And for travel plenty of countries require proof of vaccination or you don't get in the country.
Not sure if this the case yet. Any examples? But I can see it becoming the case in the future ...
A number of African countries require proof of yellow fever vaccination to enter. The vaccine certificate is issued by each country but conforms to a WHO standard.
I should have said require it for other diseases not Covid