Disney does not require vaccination for anything else, so I can't imagine this ever being a requirement.
My husband, daughter, and I all work in the medical field. The vaccine is being offered to employees, but it is not required. (And these are large hospitals where you will be suspended/fired if your flu shot or TB test are overdue)
No, there is almost no way to prove you have it that can’t be faked.
Yes, there's not a centralized vaccination record system that they could look you up in and there isn't anything "official" that you would receive as proof. Everyone would just have to bring a copy of something from their doctor's office or clinic which would be very easy to forge by changing the name or adding an extra line listing a vaccine that wasn't actually received.
Do you want them to install thermal imagers throughout the park to make sure no one is running a fever?
You say this as though it's completely ridiculous, but I thought they already were taking temperatures. I've been way out of the loop on Disney's policies and practices, so maybe that is not the case. My parents were there in September and I really thought I remembered them mentioning temperature checks.
I recently travelled to Mexico and they were taking temperatures everywhere (restaurants, parks, tours, hotel, etc). And, yes, they had thermal imagers at the airport with people monitoring them.