There's a formula on the main site for requesting an accommodation for an ESA. The issue of deposits and extra rent is kind of a grey area for ESAs. Generally speaking, a person should not be charged extra rent or a deposit for them but sometimes they do. Here's the thing. An ESA really is basically a pet. We reasonably expect a higher level of training and lower level of damage or wear and tear on a residence from an SD than a pet or ESA. It's very clear they can't charge extra for an SD, but isn't exactly spelled out that they shouldn't charge it for an ESA. If you want to make a case for waiving a deposit for an ESA, look through the webcasts over in the law section. Look for one including Herb Ziegfield (I don't have his last name quite right, but it's close). There's a good quote from him to the effect that deposits should not be required for ESAs. He's an official with HUD, so a good source to quote. You could drop that quote (with citation) into the ESA accommodation letter and see what happens.
Remember that regardless of whether the animal is an SD or ESA, the owner is still responsible for any actual damage done by the animal beyond normal wear and tear such as a human might cause. So any chewed wood work, soiled/stained carpet, etc. would have to be replaced or repaired at the renter's expense.
My personal tendency would be to replace the term ESA with "Service Dog in Training," and indicate he is in training for you. Why? There's a case, I think it is Green v Clackamas (again I got the first name right but the second is misspelled). You'll find a copy of the Green case over in the caselaw section of the main site. The Green case was about a woman who was training a hearing dog for her son and the landlord wouldn't allow her to keep the dog. Now if someone is training a service dog for an organization or for someone else, that isn't going to fly. But if they are training the dog for someone living in the household, this case could be used to support a claim that this is a reasonable accommodation because Green won her case.
The search engine I've got over there on the main site (
http://servicedogcentral.org/content) is really very nice. Just toss emotional support letter in there and you should find the sample letter easily. A few versions in fact, because I need to go do some tidying up over there. I like the ones from "Norville Rogers" best. (Norville Rogers is the real name of Scooby's friend Shaggy.) There's also a sample letter for a service dog that you might want to compare. For the caselaw and webcast stuff you could toss those terms in the search engine, or simply go to the "service dog laws" section from the main menu in the left side bar of every page and follow the links from there.