And it is legal, just as “no shirt, no shoes, no service” is legal:
Proof of a Covid-19 vaccine may be required to take part in many leisure activities next year.
By spring time, about 1/3 of the US adult population may have been vaccinated and there will be intense pressure to “open up” for the privileged vaccinated, even if this means denying access to many others who may be waiting until June-September for vaccine availability.
Large event venues (sports, concerts, events, theaters, museums) and airlines are both interested in using “immunity passports” to restrict access – and many indicate they would like to do this sooner than later. To the extent these are private businesses, they can choose to mandate vaccinations, if they wish – at any time.