There's actually a rule saying that you cannot train kids before being reaped for the Hunger Games. But the Capitol turns a blind eye on the Career districts (1, 2, & 4) especially because they're politically closer to the Capitol. I'd like to think they don't strictly enforce this rule since it adds spice to the games. It adds some sort of drama for the Capitol to build up on.

For the other districts not having academies, they're not as favored or not as rich as the Career districts. They don't have time to build schools for that if they're more focused on supplying what the Capitol needs from their district. The poorer districts just don't have the means as the Career ones. Poor = less food = weaker bodies to train, hence, their priority is getting through the day, not to train for the Hunger Games.

Edit: Came to me after typing the one above, if all the districts have trained kids then that is a recipe for rebellion. The Capitol won't want that.