As of right now, there is no vaccine passport in the USA.
But there was talk of making a government program for people to prove they are not a risk for spreading Covid either by receiving the vaccine or by having had the disease itself.
At this point, it sounds like the government is not planning to proceed with any type of "passport" program.
There were also several state Governors who said they would not allow such a program in their states.
There are HIPPA laws that this type of program would be in violation. So if it came into effect, we know it would end up in court very quickly.