Also an issue - we go anywhere pretty much without a mask if we are vaccinated. But what is to stop someone from just saying they are?
That is the idea behind the vaccine passports - an idea hardly without issues, but ostensibly the answer to your question.
There are folks out there printing bogus vaccine cards as it is, Kid. (Oddly enough, reported about previously here.)
We even know who is most likely to do that sort of bogus reporting and I will give you a hint: they are far more often associated with the right wing, politically, than the left or center.
What would you propose either states or the federal government do about it?
As for "still up to states and businesses," that is true except where interstate commerce is concerned. Trains, planes, and buses, at a minimum, are covered by the feds.