The Lincoln School surely has one. http://www.lincoln.edu.ar/
My brother played there when we attended that school. But it may be open only to current students. If so, maybe you can ask to form a non-student team to participate in their league, by gathering other kids in your same condition.
There is a "national" baseball field, on the route to Ezeiza airport. You can look it up. If always recall seeing a green sign near some exit. You could call-go there to find out what their leagues are.
Our son plays at club communicaciones. There are a few other teams around the city, depending where you live that may be one closer to you. We are in Olivos and that is the closest team to us. As BAJoe said, Lincoln has a team but you have to be a student to play there.. You can contact the "team mom", Guadalupe at email@example.com for more info. Hope to see you there.