Is it permissible for a husband to take from the food basket distributed by his institution, with the institution's knowledge of this, especially since they are too modest to ask and do not obtain such assistance from other sources?

There is no harm in the husband taking a food basket from his institution, as it is a gift from the institution to its employees. Accepting a gift is permissible for both the rich and the poor, due to the Prophet's (peace be upon him) saying: "Whoever receives a kindness from his brother, without asking for it or eagerly anticipating it, should accept it and not refuse it, for it is a provision that Allah, the Almighty, has driven to him."