There are 3 riding clubs in Buenos Aires - Club Aleman, Club Buenos Aires and Club Hipcio Argentina. Aleman & BA are located on Dorrego (btwn Libertador and Lugones) and CHA is on Alcorta, next to River.
Actually, there is a 4th - GEBA which is located directly behind Aleman but I heard that it is now part of Aleman.
All have riding lessons for children. You have to join after a few months (they are private members clubs) but don't know how it works if you have children in the riding school.
I go to a place in Cañuelas that offers lessons for kids. It is a family run club where the whole family is in the business doing something, from equitation to jumping to polo to breeding. Cañuelas is about 40 minutes from the center in car depending where you are, or 50 minutes from Congresso in a combi. I know it sounds like a bit of a trek but the place is really beautiful and tranquilo and fun with good people, so to me it is worth it. Send me a message if you are interested!
Hey there - you can try contacting this excellent teacher I met through the TOM's shoes organization... His name is Edgar Ortelli and he speaks English. You may reach him at firstname.lastname@example.org He has his own estancia where he offers lessons with very good horses.