you dont need a database.
you can instead just have smart contracts

imagine a vaccine centre signs with their keypair a message of
patient ID: <your personal publickey>
date: 12/3/21
batch: 143

that signature is proof you got vaccinated by that vaccine centre.
each signature is unique but each signature has a validation method.

you can carry around that signature and that signature can verify its contents. with a simple check
you can prove you are that patient by you signing your own message that corresponds to your public key

no database. no blocks no chains needed
airlines dont need access to all patient records. dont even need to validate all records on some database
they just ask for the signature from those wanting to fly