is there proof of vaccination that they receive?

You receive a certificate that is valid nowhere since you might have contracted COVID 19 anyways between getting the shot and showing the certificate.

Confirmation emails, certificates, stickers, ice cream? No, yes, no, only if you identify as BIPOC.

How does someone able to book a second shot? They are just making it up as they go along. Second shot is same as first. Just walk into any clinc. There is no agency keeping track of who has been given what vaccine and what safe options are for a second dose