You can google "vaccination cards" and look at a ton of images out there showing they're all filled out differently and there are no standards by which they're being completed.
All I did was find one with the information filled out like THIS and then scribbled in some jibberish under the "healthcare professional / clinic" site so it was unreadable.
Go look at all the vaccine card images and see for yourself: there is NO standard by which they're completed, they're completely useless because of that. Tracking? BWAHAHAHAHAHA! There's a term we IT Professionals have had since way, way, way back in the early 1980's when I was coming up the ranks: GIGO (Garbage In/Garbage Out.)
And that's why any database with vaccine card info. in it would be worthless.
Contract worker for California department of public health called me and asked if I was vaccinated or wanted to make an appointment to get vaccinated. I told them it was none of their business! Apparently they got a lot of similar or worse responses.