They are required if you want to immigrate to the US [1], so I sure hope citizens-by-birth also have to follow similar rules. Maybe you don't need to prove anything because by default people would already either be vaccinated, or had to obtain an exemption?
There are vaccines required to attend public school, but if your parents just say "I refuse" they are not going to kick you out. These hard-core antivaxxers are pretty small in number though.
It's one of those cases where it's required by law, but the law is never enforced.
I have applied for visa in multiple countries, attended public schools as well as university, and even had employment contracts specifying current vaccination.
I have never had to actually present any of that information. I'm not even sure how I could find it.
Interesting! So I guess the enforcement will only start when it's needed to get rid of some troublemaker. "-The person is causing us grief, what can we do to expel them? -Check their vaccination status, and look for other normally unenforced rules, there's gotta be something!"
[1] https://www.uscis.gov/policy-manual/volume-8-part-b-chapter-...