Landlords do not have the right to say no to a tenant who submits the legally required documents in order to establish an animal as a service/support animal. There are federal and state laws that regulate service and support animals.
In most cases, the landlord may require that the tenant provide a written statement from a health care professional indicating that the person is disabled and is being prescribed the animal in order to allow them to enjoy the home the same way that a non-disabled person would. A perfect example of this would be a service animal for a person who has impaired sight. Be aware that a “health care professional” does not mean only a doctor.
When a tenant meets the legal requirements to have an animal as a service/support animal then the landlord cannot charge an extra deposit. The best way to consider a service/support animal is to treat it like a wheelchair. You wouldn’t charge an extra deposit to a tenant for their wheelchair, so the same applies to the service/support animal.
If you encounter a tenant/applicant with a support/service animal and do not know how to proceed, it is best to seek legal advice from an attorney who handles landlord/tenant law and is experienced with disability accommodations.