I suppose they could. It's been decades since I've needed proof of a vaccination for travel and don't recall what I carried but vaccination cards have been around for over a century. I think the WHO card below is used mostly for yellow fever, that's not the reason for the color, but it or something similar would suffice. They don't issue it, just created the format. Not much different than an international drivers license. Though it would work, I'm not sure I want medical records in my passport.
