Is it permissible to defraud the intermediary to recover money paid to a person who was unable to fulfill a need, knowing that the intermediary possesses the means to restore the right?

If you are unable to obtain your right through legitimate means, then it is permissible for you to take it through stratagem (the issue of al-Zafar), provided that you do not wrong anyone or take more than what is rightfully yours. You must take the right from its owner, not from an intermediary, unless the intermediary has deceived you. In that case, it is permissible for you to take your right from him through stratagem.