If you're talking about personal essays and you're not a big-name trainer, rider, or breeder, I'm afraid you're going to have a hard time finding a market. I know. I've been trying for years. My best option has been anthologies. Check the Adams Media site. They just put out two horse-related anthologies and may have another in the works. Chicken Soup is a possible market as well. Several of the glossy horse mags take essays for their readers' input page, but I don't think they pay for tose.
The only exception to the personal essay rule is the essay that chronicles a specific, serious or rare veterinary or hoof problem or the "My Horse Survived Three Days in Public High School!" kind of thing. Most of the glossy mags will take those. Photos are a big plus.
I'll be curious to hear whether anyone else has additional suggestions. Thanks for posting this question.