Miniature horses actually make great service animals. According to the Miniature Horse Association, miniature horses are often between 2 and 3 feet tall and weigh between 150 and 250 pounds, so they're not that big after all. Additionally, horses are intelligent, recognize human emotions like dogs can, can be trained to stay very calm (think of those zen police horses amidst chaos), and are sometimes more accepted than service dogs.
You'd be surprised at the number of people who choose miniature horses as emotional support animals (like this guy). Another big plus is that horses live for much longer than dogs or cats (30 years is the average age), meaning that the special bond you have with them is going to last a lot longer.
The American Disabilities Act (ADA) definition of service animals is limited to dogs, although it has one exception: miniature horses that are specifically trained to be service animals. But we're talking about emotional support animals, not service animals. Emotional support animals don't require any training at all.
But in the case of miniature horses, they're relatively easy to train, are naturally inclined to follow a leader (in this case, their human) and have been living alongside humans for around 6000 years, meaning that they are pretty tuned-in to humans' behaviors and emotions. So even if they're not trained professionally, their demeanor means that they're well suited for the job.