As @Anna Sagatelova posted, use petscreening.com. The perspective tenant does all the work and I believe it is free for them if it is truly a registered support animal. Also, you are allowed to ask for the doctor's letter. You cannot ask why they need the support animal but you can request doctor documentation.
Also, just because you cannot reject an applicant because of a support animal does not mean you cannot reject them for any number of other things. Have clear established expectations and requirements for your application process and you can likely reject them for one of those items.
Why don't you want to allow for pets? I am a big fan of allowing for pets and then charging more for the property to avoid this game altogether. If they want an animal, they will find a way so you might as well make some extra money for the animal. Animals are typically a reflection of their owners so if you have properly screened a good tenant then it is likely you won't have issues.