This isn't true, at least here in the US.
At a Fortune 100 company I work for, I've been told HIPAA rules forbid them to ask directly--and they act as if that were so. They're very PC, so I'm sure they would ask while speaking through a proctoscope if they were allowed.
Employers are not prohibited from asking their employees if they have been vaccinated against COVID-19 and there are no federal laws that prohibit an employer or business from asking for proof of vaccination status. If an employer asks an employee about their vaccine status, the employee can refuse to answer, but there could well be consequences.
https://www.hipaaguide.net/can-employers-ask-employees-about-covid-19-vaccinations/