How do I claim my Emotional Support Animal on my tax return?

An emotional support animal expenses are not deductible as a medical expense.

Go to this IRS Publication 502 for medical expenses - https://www.irs.gov/pub/irs-pdf/p502.pdf

Guide Dog or Other Service Animal
You can include in medical expenses the costs of buying, training, and maintaining a guide dog or other service animal to assist a visually impaired or hearing disabled person, or a person with other physical disabilities. In general, this includes any costs, such as food, grooming, and veterinary care, incurred in maintaining the health and vitality of the service animal so that it may perform its duties.

You probably can't.

You can deduct a service animal.  A service animal is specifically trained to perform one or more assistive tasks that supports the medical needs of the owner.   If you have a service animal, you can deduct certain costs as outlined above.

Generally, an emotional support animal is different from a service animal in that an emotional support animal provides support through its own natural behaviors, and is not trained to perform any specific type of medical assistance.  Emotional support animals are not deductible medical expenses under current law.