I don’t think this is a good idea, or necessary, for many of the reasons shared here already. In my experience, this hasn’t been an issue. In the 6 US sits we have booked since vaccines were available to all, not one owner has asked us about this, and because my husband and aren’t worried about their status, we don’t ask.
I know some of these people have gotten the vaccine for sure themselves based on things they said when meeting them ,or where they were planning to visit, so one thing that is clear to me is not assuming owners who don’t make mention of it must not be, or that anyone who has gotten it wants the same of the sitter.
If sitters see an advantage to publicizing this, they can do so on their profiles and application messages, and any owner who requires it can put it in their listing. Yes, I’m sure it is annoying when you get messages from people who clearly didn’t read your listing and all the ‘deal breakers’ but such is life.
This doesn’t pertain so much to this particular issue, but in general, I think people sometimes expect too much of the site, and want them to do things it is not on them to do.