Hi Mary,
Cameron County is the Brownsville/South Padre Island/Harlingen area. There are stables/clubs and stuff in the Los Fresnos area which is just north of Brownsville. If you search through this forum, someone once had the same question (or close to the same) that you had and they found some luck in Los Fresnos area. Oh here it is:
Farasi Farm
Sorry, I just know of that one ...but I am sure there are more in the area. We used to own a horse (neighbors thought it was a big dog) and had to eventually give it away as housing development began to engulf our area.
Good luck!