Vaccination cards? In which century do you think we're living? We got apps for that.. with QR codes that can be scanned, to be double checked against government databases. Which can be updated, should a booster shot become necessary (and show validation dates, which can be shifted, if there is more knowledge about how long vaccinations stay powerful enough, or new variants require booster shots sooner)

In fact, we had plenty of issues with Americans bringing their "vaccination cards", which looked so amateurish, and couldn't be verified at all, when we opened our borders to you much earlier.