Go ahead. That sounds fine. The academy is also classified as a philanthropist foundation. That is to say, non-profit. It's funded mostly by donations and internal interest. (That is to say, some staff pay out of their own pocket from their side jobs.)
I'd say that your character would likely be much more concerned about passing the exams to get in than having the money. As I mentioned before, they'd find a way to make it work. That being said, the academy isn't exactly a big spender. Most of what the academy gives a student is the buildings and teachers themselves, with food and pokecenter care being also notable. Anything in Oak Town for instance such as TMs, items in general really, non-cafeteria food, movies, all that stuff is not covered by the academy.
Hence why I suggested he'd probably have to have a guardian of some sort, even if it was one assigned to him.
Am I able to use Karen of the Indigo Elite Four briefly in my character's history? It wouldn't be anything major, just her being my character's inspiration for becoming a pokemon trainer.
And secondly, are there anyways for someone without a lot of money to get some sort of scholarship through a competition or something? Because I don't exactly think that an orphan would be able to afford it. Unless it doesn't cost anything (but I doubt that's true).
It's a very free form RP. The plot is what you make of it, is what I'd say. There are certain things I throw your way like two classes at the beginning of three days and the occasional rocket plot, but...