The provision you cites says nothing of the sort. You are misreading it.
Unless there is a State or Federal law that mandates employers must require records of inoculation for terms of employment, your PHI (Private Health Information) cannot be required under HIPAA law. You can freely give this information, but they cannot require it. Your private health information is absolutely protected!
Of interest:
PHI includes health records, health histories, lab test results, and medical bills. Essentially, all health information is considered PHI when it includes individual identifiers. Demographic information is also considered PHI under HIPAA Rules, as are many common identifiers such as patient names
https://www.hipaajournal.com/considered-phi-hipaa/