Is it permissible to enter a school through an intermediary, knowing that entry is secured through an intermediary and the required donations?

It is forbidden to use intercession (wasta) to be exempted from tuition fees in public schools if you are obligated to pay them and are not eligible for the exemption. It is permissible if the school's regulations allow it. As for private schools, the matter is easier, and it is permissible to use intercession to obtain an exemption.