Is it permissible for his wife to take a loan from the Bab Rizq program in order to give him the money?
There is nothing wrong with taking loans that do not involve paying any interest.
If your wife takes a loan, she may give it to you by way of a loan, or as a gift or partnership. But if she takes a loan in order to work on a specific project, then she has to fulfil that, because the Prophet (blessings and peace of Allah be upon him) said: “The Muslims are bound by their conditions.” Narrated by Abu Dawood (3594); classed as saheeh by al-Albaani in Saheeh Abi Dawood.
It is no secret that it is not permissible to lie, deceive or use tricks; rather it is obligatory to adhere to the terms of the contract between your wife and the lender.
And Allah knows best.