There is no "how many". Unlike a traditional university that give out scholarships and have red shirt freshmen, and a set roster, the academies are a little different. Except for a small handful of freshmen, the rest of the freshmen are on what is called a jv team. They aren't on the regular varsity roster. They do play games, every Friday, but they are against junior colleges and d2 schools. But the practice and do everything else with the varsity team.
So, the academy can and will take any walk ons they ant. Then, by the end of the first season, they will get rid of "x" amount of players. Most freshmen, plus som others. Then, at the end of spring training, thy will get rid of some more. Then, at the end of summer practice, which the freshmen are now sophomores and a new batch of freshmen are coming in, the will reduce more to get to the NCAA roster level they are allowed.
For what it's worth, there are between 50-60 freshmen coming in each year. By the end of the first season, there will be about 30-40. By the end of spring training, there will be 20-30. In the end, by the time the freshmen ar seniors, there will be around 15-20 left. This season, there are 18 seniors. Most that are cut, are done in the first year. Some are cut in years 2,3, and 4. Some quit on their own because their interests have changed or they find they can't do both football and the academics. Same with all sports. When my son was a freshman and played football, a handful of freshmen, who were recruited just like him, didn't even make it through summer practice after basic training. They knew it was going to be tough and quit football right after BCT and before the season had already started. And for what it's worth, to those who question recruited athletes, these individuals, while recruited athletes, were direct entries, didn't go to prep school, and had excellent applications in their own right. They just happen to be able to play football besides being academically proficient.
So, to answer your question, the academy can take as many walk ons or recruited athletes as they want to. Whether they get to stay on the team more than a month, summer, first season, spring season, or into the next year season, is totally up to their performance and the coach. They still can only have so many on the active roster. Mind you, this is for football. Each sport has their own roster rules.