Yeah but there's no registry, there's no requirement, there's like nothing those papers prove. You don't need papers. You can't prove something that doesn't exist in the first place.
Only the misinformed business owner would require it and only the misinformed (and I don't mean that rudely) owner would think that is needed. Which is why I get the OPs question regarding documents.
Here's the information from the government themselves and pay attention to the second box:
CERTIFICATION AND REGISTRATION
Q17. Does the ADA require that service animals be certified as service animals?
A. No. Covered entities may not require documentation, such as proof that the animal has been certified, trained, or licensed as a service animal, as a condition for entry.
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Now if Biltmore wanted to refuse a service animal on the basis of their presence in their environment presents a hazard or if the service animal themselves has behavior that presents a hazard that is okay for Biltmore to deny access. But they cannot demand paperwork that which they are not allowed to ask for and paperwork that which doesn't actually exist and deny based on that.