Children are permitted to perform Hac and enter ihram according to the consensus of scholars from all four schools of jurisprudence. Hz. Peygamber (sallallahu aleyhi ve sellem) affirmed this when a woman lifted her child during Hac and asked: 'Is there Hac for this one?' He replied: 'Yes, and you will have a reward.' This hadith establishes both the validity of a child's Hac 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 tavaf 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 Hac performed by a child — whether a discerning minor (mumayyiz) or a non-discerning infant — does not fulfill the obligation of the Hac of Islam. All four schools agree that once the child reaches puberty, they must perform Hac again if they meet the conditions of obligation (ability, means, etc.). Sheikh Ibn Uthaymeen noted that while the child's Hac is a valid voluntary act of worship with reward, the farz Hac can only be fulfilled by someone who has reached the age of legal responsibility (bulugh).