Is it permissible for the wife to apply for a loan in her name for a small project from the Abdul Latif Jameel Fund (Bab Rizq) and receive the amount, so that the husband can take it and use it for his own project, knowing their dire need for the amount?

There is no harm in taking a interest-free loan (Qard Hasan (definition)). If the wife takes such a loan, she may give you the money as a loan, a gift, or as a partnership, unless she took it for a specific project, in which case she is obligated to fulfill that, due to the Prophet's (peace be upon him) saying: "Muslims are bound by their conditions." It is obligatory to adhere to the agreement between the wife and the lending institution, and not to lie, cheat, or deceive.