I don't believe that there is a specific form or request process. You can put in your listing details and house rules that you cannot accept service animals for whatever is your specified reason, e.g., you, family members or staff are allergic to animal fur and hair, you have a dog on site that's not friendly with other dogs, your floors are cleaned with chemicals that may affect dogs, etc.
If you read the policy for service animals on Airbnb you will see that you can only be exempting from hosting service animals if you are in a home share situation where you have your own pets or if you have pet hair allergies for example if you are asthmatic. @Karla420