I see no reason why a grocery store couldn't require a vaccine to enter. The badge is irrelevant really since it is a privately owned business and businesses put rules in place to enter their businesses all the time. A business could charge a fee to enter and issue a badge if they wanted... just from a business standpoint that would be a bad decision. (Unless all the businesses started doing it... but let's not give them any ideas...)I was speaking solely to the fact that right now, they are EUA vaccines which is a pretty big gray area as to what is legally allowed under a EUA for requirements. (I sure don't have those answers).
I'm not sure how a court would rule on it, but I think the distinction matters. Someone needs a necessity (food) and i think a court might find a vaccine restriction unreasonable for necessities of daily living.
Alternatively you dont have to go to gencon or use air travel. These require a purchase along with compliance with their rules. As long as it isnt discrimination they can set health protections for attendees.