Both of these screeds are from leftists who are whining about the impossibility of verifying vaccination, but they are accurate in so far as what the system is:
https://www.theatlantic.com/health/archive/2021/05/america-covid-vaccine-honor-system/618891/
Even NY’s system doesn’t prevent people from sharing screenshots and it only can track NY shots and then only those who haven’t opted out of the State database. (Something I recommend everyone doing if you are in an Opt out state.)
Depends on the State, then. California has a mandatory IIS in place to track all vaccinations, and I know that the digital record exists for my COVID-19 shots, because I pulled my own record. The digital record matches the information on the card, and has a QR code.