Is it permissible to pay a sum of money to an intermediary who acts as a guide and mediator between a buyer and the director of an institution for the sale of a dwelling, in order to facilitate the purchase process which cannot be completed except through him, knowing that he cannot request money from the selling institution?

We understand from the question that the dwellings are sold at a specific price to those who meet certain conditions stipulated by the state. The mentioned mediation is not a brokerage but rather an intercession (shafa'ah), and it is forbidden if it enables you to purchase the dwelling without meeting the required conditions. It is permissible if you are eligible for the conditions but can only obtain your right through intercession. Regarding the ruling on the intercessor taking money in exchange for intercession, there is a difference of opinion among scholars, and the preponderant view is that it is permissible if the intercession was not obligatory upon him.