My wife and I got vaccinated at two different CVS in Texas. The only real proof is the little card with handwritten info on it, as many said.
When I got shot at CVS, it asked my email, but didn’t ask my wife’s when she got hers. I got an email vaccination record, but not my wife. And I don’t see my record on CVS website, at this moment.
As I see it, they just want to get people vaccinated as soon as possible. They didn’t have time to think through the whole thing. “Keeping the record” is a low priority. It’s a mess, even in the USA.
Serious as Covid, countries will need real proof of no virus before a traveller can enter. Either a piece of paper or a printed record has no credibility, at this moment.