I didn't think you could charge a deposit for an ESA.
+1. ESAs are not animals or pets, legally, so you cannot charge them pet deposits or rent. You can attempt to charge them cleaning or damage fees when they leave but the kind of people who buy a $25 ESA certification off the internet are not the type to give a damn when they leave.
Fortunately OP is living in his 4-unit and doesn't have to abide by HUD laws and so may restrict because of animals (support or otherwise), age, sex, household size, etc.
Luckily, I've only had two ESA applicants. Both had other issues that disqualified them (credit, income, prior rental history, criminal history) and this seems to be the safest way to disqualify someone without risking a HUD complaint.
Other avenues to decline an applicant with a ESA(s) are 1) insurance and 2) doctor verification.
1) If your insurance does not cover the animal in question (dog breed restriction, exotic animals, etc), it is an "unreasonable accommodation" for the property owner to have to change insurance providers or pay for a significantly more expensive policy to accommodate the ESA.
Source.
2) You cannot ask the applicant about their condition or disability but you can ask for verification of the disability if it's not obvious (ie: a blind person's disability would be obvious; a veteran's PTSD would not be). The verification would be a letter from a mental health professional. You then contact to the professional to verify that they did write the letter (outright falsification is common).
This is third hand knowledge but supposedly a very effective way of dealing with fraudulent online certifications is to contact the website/doctor on the letter, let them know that if the ESA damages property or injures a person, they would be legal and financial responsible for the damages if it is found that the certification/verification was issued fraudulently. Again, this is third hand knowledge, but supposedly the online cert factory people rescinded their certifications every time when told this.