Getting a covid vaccine has been a personal choice, up until now. So there's a lot of information here, lets take it piece by piece.
Q: Should you let a potential employer know that you have been vaccinated?
A: Absolutely. Especially if you're going to be working in someone's home. (ie a dog walker, nanny, handyman, etc)
Q: Can an employer decide to hire or not hire me based on whether or not I've been vaccinated?
A: Legally, yes they can. As long as they can prove that it's necessary and job related.
Q: Can my current employer ask if I'm vaccinated?
A: Yes they can and they ask for proof. Here's where it gets dicey though. If the employee has NOT been vaccinated, they cannot pry as to why, as it might cross the line into privacy issues, especially when the shot is not required for the job.
Q: If I'm not vaccinated, can my employer require me to get the shot?
A: Possibly yes. They just have to prove that it is a legitimate job requirement and they can make accommodations for those who cannot get it because of a disability or religious reasons
Of course states can pass laws at any time to make any of this legal or illegal. So basically our constantly changing info on Covid is still constantly changing.