No, not really... if someone supplies “proof” there would need to be a way to check it. That would require someone along the line to share that medical info - so not only would you need to supply proof, you’d need to sign a new HIPAA waver with the owner of that info so that it could be shared. I have serious doubts about anyone wanting to go down that path.
In any event, when businesses start requiring proof of flu, measles, and other vaccines, then I will believe they’ll require proof for Covid. Thus far only places like schools require proof of vaccination and they have exemptions for medical and religious reasons - and the “proof” of those exemptions is a simple form signed by the parent/person. I exempted my kids from varicella vaccine and HepB (var. they now have, HepB they will get at 16, and yes their pediatrician agrees with my reasoning) so I know they don’t require a note from a doctor or pastor or something. Proof of covid vaccine would also need exemptions, and again I doubt they would require anything official - it would simply be too onerous to check the veracity, and heck WDW can’t even ask for proof of disability to get a
DAS pass due to privacy laws attached to the ADA - reasons for not getting a covid vaccine could easily fall under that.