Carrying a child during tawaf is boleh and does not invalidate the tawaf of the carrier according to the consensus of scholars. Nabi (shallallahu alaihi wa sallam) himself prayed while carrying his granddaughter Umamah bint Zaynab, setting her down when he prostrated and picking her up when he stood. This establishes that carrying a child during acts of worship does not invalidate them. Furthermore, Nabi performed tawaf while riding his camel, demonstrating that tawaf is valid even when not walking on one's own feet.
Sheikh Ibn Baz ruled that if a person carries a child during tawaf and intends the tawaf for both themselves and the child, it counts for both of them. He based this on the hadith of the woman who asked Nabi whether her child could perform Haji, and he confirmed that the child would receive reward. The guardian performs the rites on behalf of the non-discerning child, and tawaf performed while carrying the child fulfills the tawaf requirement for both.
Sheikh Ibn Uthaymeen added that using a baby carrier, stroller, or having someone else push the child in a wheelchair alongside the person performing tawaf are all acceptable methods. He noted that the key requirement is that the carrier completes seven full circuits around the Kabah. The practical concern is the extreme crowding during peak times, and scholars recommend choosing less crowded times or performing tawaf on upper floors when carrying young children for their safety.