A minor cannot perform proxy Hac 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 farz Hac. A child's Hac, while valid and rewarded, is considered voluntary (nafl) and does not count as their farz Hac. Since the child has not fulfilled their own obligation, they cannot fulfill it for someone else.
Sheikh Ibn Baz stated that the proxy for Hac must be a mature Muslim (baligh) who has completed their own farz Hac. He cited the hadith of Shubrumah, where Hz. Peygamber told a man to perform Hac for himself first before performing it for Shubrumah. If an adult cannot perform proxy Hac 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 Hac. A child may perform Hac for themselves (as a voluntary act with reward), but they cannot serve as a representative for someone else's farz worship. Parents looking to send a proxy for a deceased relative should choose an adult who meets all the conditions of proxy Hac.