Children are permitted to perform Hajj and enter ihram according to the consensus of scholars from all four schools of jurisprudence. The Prophet (peace be upon him) affirmed this when a woman lifted her child during Hajj and asked: 'Is there Hajj for this one?' He replied: 'Yes, and you will have a reward.' This hadith establishes both the validity of a child's Hajj and the reward earned by the parent or guardian who facilitates it.
For infants and very young children who cannot understand or act for themselves, the guardian (parent or wali) enters ihram on their behalf by making the intention (niyyah) and pronouncing the talbiyah for them. The guardian then performs tawaf and sa'i while carrying the child, and throws the pebbles on the child's behalf during the stoning ritual. Sheikh Ibn Baz explained that the guardian should ensure the child wears ihram garments if possible and avoids ihram restrictions as much as practically feasible.
However, the Hajj performed by a child — whether a discerning minor (mumayyiz) or a non-discerning infant — does not fulfill the obligation of the Hajj of Islam. All four schools agree that once the child reaches puberty, they must perform Hajj again if they meet the conditions of obligation (ability, means, etc.). Sheikh Ibn Uthaymeen noted that while the child's Hajj is a valid voluntary act of worship with reward, the obligatory Hajj can only be fulfilled by someone who has reached the age of legal responsibility (bulugh).