Employers need to know if their employees represent a risk to the health of their employees and customers. Nobody wants to employ Typhoid Mary.

We should be extremely wary of giving employers our entire medical history, which is illegal for employers to collect in the US as well (under the ADA and other laws). We should have no concerns giving them our vaccination history for at least two reasons: 1) that info is already held by the government and we’ve already given it to K-12 schools plus university. And 2) as sensible members of society, we are vaccinated so there’s nothing for our employers to act against us on.

I suspect in all the cases where companies require vaccines, it will just be the company asking people, not inspecting medical records. I mean, given all the details about the reality of society, doesn't that seem like a more probable hypothesis?

it's unlikely there will be any large changes to private health data laws at the federal level of US in the near future. The trend appears to be that states will do this instead.