Nope - would play out just like the GAC and service animal stuff. The gain people see far outweighs any consequences (or in this case.. effort) so selfish people will quickly flock to it.
Look at how quickly the stupid fake 'mask exception' stuff spread. This would be similar. Where there is no consequence for cheating, large portions of our population will gravitate to it because they are selfish.
Creating and passing fake IDs is an actual crime - so there is a fear of consequence. Additionally, you can't just fill out a form, click print, and get a passable fake ID without a bit more effort... unlike a vaccination card. Plus, those IDs have reasonable levels of expecting what it should look like ways to pass at least basic scrutiny. It's not really the same thing.
This is more on the level of 'requiring a doctors note' - or 'a note from your mother' - There is zero way for a business to actually scrutinize proof people provide. It's no more effective than 'just tell me if you are vaccinated'.