Is it permissible for my husband to appoint someone to fulfill the vow he made to teach the needy for a full year for the sake of Allah, or is he obligated to fulfill it himself in the country where he currently resides, or should he wait until he returns to his homeland?

The Prophet, peace and blessings be upon him, forbade making vows (nadhr), but if one vows an act of obedience, it becomes obligatory to fulfill it. Therefore, your husband must fulfill his vow. It is permissible for someone else to teach on his behalf if he did not intend to teach directly himself, as long as this deputization serves the intended benefit. He may also teach needy students in the country he traveled to, unless his vow was specific to a certain country. It is permissible to delay fulfilling the vow until his return to his country, unless the wording of the vow necessitates immediate fulfillment.