pay attention to your state law.

1. federal law only requires you provide reasonable accomodation for disabled people, including service animal for disabled.   If your tenant has no disability (including mental illness), refusing the animal does not break federal fair housing law. 

2. state laws varies, see 

https://www.animallaw.info/topic/table-state-assistance-animal-laws 

If your state requires you to accept emotional support animal, you cannot charge pet rent or pet deposit.