Under the Fair Housing Act (FHA), an emotional support animal is a form of reasonable accommodation of disability. If you want to take action that would otherwise violate your lease, it is best to get the letter in advance, show it to your landlord, and explain that you're bringing the animal into the home as an ESA pursuant to the landlord's obligation to reasonably accommodate your disability under Sec. 804 of the FHA.
An ESA is not a service animal. Service animals are protected by the FHA and the ADA. ESAs are not covered by the ADA. You have no right to take an ESA into a grocery store. (ESAs are allowed on airplanes under a separate law, the Air Carrier Access Act.)

