Step 1, ensure it is a service animal, not an emotional support animal. You can ask two questions. Is it a service animal and what the animal is trained to do. Per the ADA only dogs and small horses can be used as service animals, so if someone tries to tell you they have a service monkey it is BS. Emotional support for animals DO NOT have the same legal protections as service animals. The only benefits an emotional support animal has is being able to live in normally pet free housing and travel in the cabin of an aircraft. They are not allowed in places that don't allow animals like a service dog is.
Just like a business we have the right to refuse access to the service dog if its behavior is aggressive or its behavior is not controlled.
Step 2, ask the patient what they want you to do with the animal, they probably have a back up plan.
As a general rule the dog shouldn't be separated from the pt, however most ambs do not have a way to secure the dog in the back. If the dog is small and has a kennel, you might be able to secure the kennel somehow. Theoretically if the dog is big enough you can buckle them into the front seat.
A lot of service dogs wear harnesses and people commonly buckle the seat belt through the harness. If it is a small horse, you are totally on your own. Luckily they aren't very common.
The exception to trying to keep the pt and dog together is that if the pt is something like a cardiac or stroke activation, and they aren't going to be in the ED long before going to the OR. The animal will not be allowed in the OR, service dog or not, and you don't want it at the hospital without someone keeping an eye on it. In that case it is best to have the dog cared for by someone else, friend, family, pastor...whoever you can get a hold of. I would only call animal control as a VERY last ditch option.
If it is something like an MVC and the dog is also hurt animal control may be the only option to get the dog to an emergency vet, but I would be appealing to our supervisor, the bat chief, or PD to take the dog in before I went to animal control.