I mean, from a simple enforcement perspective "prove that you're vaxxed" runs into the same problem as "prove that you're a legal resident".

Access to health care, access to documentation of that health care, and the ability to produce it on demand all require certain amenities that marginalized people don't have. It's a rule that inevitably penalizes people for being poor.

Shy of getting people chipped and slotting your medical records into the same system that we use for criminal enforcement, the folks enforcing the laws will default to the assumption that you're at fault until you can prove otherwise.