What is the ruling on benefiting from the services of a non-profit Christian organization that provides housing assistance to those in need (Muslims and non-Muslims) under certain conditions?

There is no impediment to a Muslim accepting aid or a gift from a disbeliever. The Prophet, peace and blessings be upon him, accepted gifts from disbelievers. There is no harm in accepting permissible aid from organizations, provided that it is not in exchange for compromising one's religion or cooperating in falsehood. Caution must be exercised against inclining the heart towards them and showing them affection.