There is a case for legitimate and trained animal assistants, mostly dogs.
You can buy a "Support Animal" vest online. People come into my museum with untrained, yappy dogs with a $5.99 vest. Boss says we can't turn them away. Some have urinated or defecated in the exhibit areas.
You can turn them away. This is a myth, and if more businesses would do so this nonsense would stop.
Only Service Animals have protection under the ADA. These are animals that are trained to perform a specific task for people with a covered disability. Emotional Support animals are not covered have exactly zero rights under the ADA, but States might allow them under State Law.
The only questions you are allowed to ask are:
(1) is the dog a service animal required because of a disability, and (2) what work or task has the dog been trained to perform.
You can't require documentation for the dog or the disability. You can't ask what the disability is.
True Service Animals (99% Dogs) are easy to spot. Well trained, welded to their owner, well behaved and not easily distracted.
Businesses act like if they violate the ADA they will be taken out and shot. You might get a fine ($1500) if you violate the law and actually deny entry to a legitimate Servce Animal. Anything else gets an "education" letter.
C'mon. Take a gamble. My favorite tactic is to ask for documentation for the dog. I think that's a loophole. You can't require documentation, but it doesn't say you can't ask for it. If they give you documentation (and many have it, they got it with their vests as it's free on those sites) it's FAKE. There is no system anywhere in the U.S. to certify and document Service Animals.
When they pull out their documentation I deny them. When they start threatening me I inform them that the penalty for claiming a fraudulent Service Animal is the same as for violating the ADA and that I'll take my chances. I then whip out my own Service Animal ID that has my picture on it and list me as a 3 year old Bichon Frise named Mr. Whipple. I'm an Epilepsy Alert dog.
If that doesn't work, then you observe. The dog must be kept on a leash at all times. It has to stay with its owner. My enforcement of this was in connection with overnight stays so we had more time to surveil them. Fake Service Animals piss me off. They make it harder for people with real ones.
But it seems most businesses don't want to put any effort into this.