I'm really shocked at how many people here who are landlords would turn someone away for this, especially when the tenant furnished a doctors note.
You're in a service industry. Follow the freaking laws. If someone is exploiting a law, it's not your job to be a vigilantly. Go through the proper channels to get the law fixed if you don't like it. You are absolutely not this person's healthcare provider and in fact, the healthcare provider told you she has it for medical need. So are you going to deny them and emotional support animal because you feel like they don't disserve it? Is that your medical call to make?
I have a psychiatric service dog for combat related PTSD and I'm also super athletic, because, you know, exercise makes life better when you're struggling mentally. I can't tell you how much **** I've caught in public, and how embarrassing it was, because someone thought that it'd be a good idea to tell me how I didn't need a service dog, I didn't look disabled, and how I was taking advantage of the system.
Imagine sitting down to dinner with family or someone you're just meeting for the first time and then having to defend yourself to a restaurant manager on what your mental diagnosis is and why you have a service dog. This wasn't an uncommon occurrence either. It happened so frequently, and was so embarrassing on a regular basis, THAT I STOPPED USING MY SERVICE DOG.
Tell me how a landlord passing judgement, on someone who has a doctors note, saying that they don't believe they need it, is any different.
Now I know someone is going to say "Well a service dog is different than an emotional support animal" and the thing is, it's not. Maybe to the government and doctors it's different, but to you, the landlord, the NOT medical professional, it's a medical device that has been prescribed to this person. It's no different than a wheelchair or a pair of crutches in your eyes.