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