A minor cannot perform proxy حج on behalf of another person according to the consensus of scholars. This is because one of the essential conditions for a valid proxy is that the proxy must have already performed their own واجب حج. A child's حج, while valid and rewarded, is considered voluntary (nafl) and does not count as their واجب حج. Since the child has not fulfilled their own obligation, they cannot fulfill it for someone else.
Sheikh Ibn Baz stated that the proxy for حج must be a mature Muslim (baligh) who has completed their own واجب حج. He cited the hadith of Shubrumah, where نبی کریم told a man to perform حج for himself first before performing it for Shubrumah. If an adult cannot perform proxy حج until they have done their own, then a child — who has not even reached the age of obligation — certainly cannot serve as a proxy.
Sheikh Al-Fawzan confirmed that maturity (bulugh) is a condition for the proxy, as is having previously completed one's own حج. A child may perform حج for themselves (as a voluntary act with reward), but they cannot serve as a representative for someone else's واجب worship. Parents looking to send a proxy for a deceased relative should choose an adult who meets all the conditions of proxy حج.