I suspect it'll be kinda organic. Once enough people have the vaccine you'll start to see airlines, trains, hotels, sporting events, concerts, some restaurants, parties, parades, cruise ships, and the like requiring proof of the shot from their customers. The Gov won't have to mandate it, private companies will do it for them from a mix of actual belief in the shot, virtue signaling, and fear of liability lawsuits.
You won't HAVE to get one, it'll just be annoying to do anything without one.
ETA: once final FDA approval is received I could see employers mandating it for any job that deals with the public as well. Just to avoid the liability of "your employee gave me COVID!!!"