Unfortunately it seems HIPPA does not provide much protection. Hopefully the pandemic is declared over before any COVID vaccines get full FDA approval because that will likely lead to some companies making the vaccine a requirement for employment.
Even if HIPAA doesn't, there are precedent cases about disclosing/exposing sensitive data about an individual, and especially health data. I already asked several HR departments on the private side and OPM (the govt equivalent of HR). They all said for now they are staying out of the vaccine verification issue, and hoping there will be executive order or some court case that would clarify exactly what can be demanded from employees in terms of vaccination status. Again, asking people for their vaccination status (especially publicly, as the example earlier mentioned) is akin to asking people publicly about their STD status, and I am pretty sure employers have been sued successfully over that, which is why HR departments do not ask such questions even though most HR departments would love to know if somebody is carrying an STD that can easily be transmitted in an office environment (without sexual contact - e.g. gonorrhea).
Here is a (verified) example. Apparently, it is illegal for a person to disclose to others the fact that some other person is a convicted felon even if the criminal record of said felon is public information. The courts have ruled that while the record may be public, the act of making an effort to dig it out and broadcasting such information unduly burdens the felon because it limits his/her employment/social opportunities and it is the responsibility of each individual who interacts with the felon to verify if the felon has a record or not. Even sex offenders are protected, even though most states require them to notify the state whenever they change their address and where their residence is. I bet we will see some lawsuits about the vaccine verification this year so time will tell I guess...