This is a point of legitimate difference. The majority (Mālikī, Shāfiʿī, and Ḥanbalī) hold that zakat is due on the wealth of a minor or an insane person and must be paid by their guardian, because zakat is a right attached to the wealth itself, and the guardian trades and manages that wealth so it is not gradually consumed. The Ḥanafīs hold it is not obligatory on their monetary wealth (though ʿushr on crops and Zakat al-Fitr still apply), since they see zakat as an act of worship requiring legal responsibility (taklīf). The stronger and safer view is that of the majority: pay it from the child's or incapacitated person's wealth to protect the rights of the poor.
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.