How can my wife, a stay-at-home mother with no work outside the home other than household chores, give charity? And is it permissible for me to give charity on her behalf?

Voluntary charity is not obligatory, and whoever does not perform it will not be punished for its omission. A woman may give charity from her own money, and she is permitted to give charity from her husband’s money for what is customarily tolerated, due to the saying of the Prophet, peace and blessings be upon him: "When a woman gives in charity from her husband's food, without causing spoilage, she will have her reward, and her husband will have his for what he earned, and the storekeeper will have a similar reward." It is not permissible for the husband to pay the obligatory Zakat to his wife, because her maintenance is obligatory upon him. As for voluntary charity, there is no impediment to giving it to her.