Is it permissible to take a percentage ranging from 10% to 20%, or even 50%, of the income of team members in exchange for supervision, referrals, and services provided? Is this percentage considered exploitation of their need? And what is the correct, Sharia-compliant percentage in this case?

Taking a commission from both parties to a contract or from one of them is, in principle, permissible; this is known as brokerage (samsarah). It is considered a form of permissible ju’alah (stipulated reward), provided that the one from whom it is taken is aware of it. If the questioner is an agent for the institution in recruiting and supervising members, then it is not permissible to take a commission from the members except with the company's permission as well. There is no specific percentage for a permissible commission defined in Islamic law.