In theory they could own the business as long as the profit of that business were used to further the non-profit work of the school.

The bigger question would be whether the school actually owns the business or whether you as a separate non-profit were operating within the facilities of the school.

If it's a private school. Yes, it can.

But a public school (read: relies on government funding) is legislatively prohibited from owning a business that may lead to an exploitation of students.

However a course like this will have no problem being run to a private or public school as part of the curriculum. A lot of schools teach something similar in a "Small Business" class where students are to go start a business and the success of the business is their grade. The problem begins if you try to make it extra-curricular. In short, the school will likely not allow it.

Great idea, I like the fact that you want to help students to grow. I can recommend all students a great blog post https://www.jpost.com/special-content/the-best-essay-writing-services-online-in-2021-672835, which has great educational resources to help you with your studies. You will definitely want a service that's capable of providing an excellent level of support, and one which can easily satisfy your needs with no hassles.

The logistics of schools owning a business as a legal entity has some legislative implications. This is a technical question for lawyers, as I do believe there are ways to navigate around it (like certain percentage of ownership...etc)

Overall, the idea is interesting, and could be worth to explore. One suggestion I have is to keep the main focus on high schools, associate degree schools (2yr colleges) and actual higher education institutions, as the dedication, enthusiasm and success rate potentially is going to be higher associated with such entities.
Good luck,
Sincerely,
KT