A minor cannot perform proxy الحج on behalf of another person عند 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 الحج. A child's الحج, while valid and rewarded, is considered voluntary (nafl) and does not count as their واجب الحج. Since the child has not fulfilled their own obligation, they cannot fulfill it for someone else.
الشيخ ابن باز stated that the proxy for الحج must be a mature Muslim (baligh) who has completed their own واجب الحج. He cited the hadith of Shubrumah, where النبي told a man to perform الحج for himself first before performing it for Shubrumah. If an adult cannot perform proxy الحج 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 الفوزان confirmed that maturity (bulugh) is a condition for the proxy, as is having previously completed one's own الحج. A child may perform الحج 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 الحج.