Is it permissible to use mediation (wāsiṭah) to obtain a delayed promotion, knowing that there is a colleague who also has delayed seniority, and that mediation has become essential to secure the right to promotion in the work environment?

It is permissible to intercede to obtain a promotion if it does not result in depriving someone more deserving. However, if it results in depriving someone more deserving, then it is forbidden. If the intercession is gratuitous, it is a good intercession (shafa'ah hasanah). As for whether it is for money, if the intermediary has no relation to promotions, then this falls under the category of "thaman al-jah" (price for one's position/influence), which is permissible according to many scholars. However, if he is responsible for promotions, then what he takes is a forbidden bribe for him, but it is permissible for you to pay it if you cannot obtain your right otherwise.