This is a simple list, the problem can be solved with the simplest of datastores.
Someone gets the vaccine from an approved source, source enters the government id of the patient and the serial number of the dose into a webform. To check if person is vaccinated compare Id.
Most, if not all, pitfalls this has (requires ID to get vaccinated which not all people have, people can fake ids, unscrupulous provider could fake doses and resell) exist in meatspace, and not because of a lack of blockchain.
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This is a people problem just like mask wearing and social distancing are. It's the reason we've seen Dr. Fauci start increasing percent of immunization we need to see before we consider herd immunity having been achieved based on how willing people are to vaccinate https://www.reuters.com/article/health-coronavirus-usa/fauci...