Jesus, google "Is the COVID vaccine protected under HIPAA." Spoiler Alert: It isn't.
But you are right. It's said how many people don't even know their rights.
Let me preface this by saying I believe everyone that can should have gotten vaccinated. I also think it's fair game to ask athletes their vaccination status.
But let's talk about what HIPAA covers and doesn't cover. It covers or prevents a covered entity, which is basically just doctors, hospitals, insurers, and those providing health care services to you, from disclosing your medical records to third parties without your consent except in limited circumstances. Vaccination status is certainly a medical record.
HIPAA does not prevent anyone from asking you about your health, nor does it place any restriction on anyone that is not a covered entity (or their business associate). So, if I shared my vaccine status with Schefter, he could share it freely. If I told my doctor, he is prevented from disclosing it. I am not subject to HIPAA, except the rights it grants to me in my personal health information.
Your employer is (most likely) not a covered entity either, nor would an NFL organization, except for the parts that provide medical services. And in, any event, NFL teams certainly require waivers of this to disclose injuries etc. pursuant to NFL rule.
So, in sum, HIPAA only applies to your medical providers. Not to you or your friends or the media or your employer. Your employer is free to enact whatever policies they deem fit, subject to laws.