this isn't really an answer, more some musing;
seeing as we don't really encounter any scholarships in the books, it's hard to say whether they exist or not. It's possibly more likely for when attending post-school education, such as a med school, that maybe if they did super well while at their school, they might be able to get a scholarship.
Alternatively you could maybe have them supported by a benefactor of some kind (possibly even an anonymous one).
Or simply working their way through with a part time job(s).