Yes, children of any age can perform Umrah, and their Umrah is valid as a voluntary (nafl) act of worship. The Prophet (peace be upon him) confirmed this when a woman presented her child asking about Hajj eligibility, and he said: 'Yes, and you will have the reward.' The same principle applies to Umrah.
For infants and very young children who cannot perform the rites independently, the parent or guardian acts on their behalf: the parent makes the intention (niyyah) for the child, carries them during Tawaf, walks with them during Sa'i, and performs the hair trimming. The child should be in a state of cleanliness (clean diaper, clean clothing). For older children who understand the basic actions, they perform the rites themselves with parental guidance — making their own intention, walking in Tawaf and Sa'i (with assistance if needed), and having their hair trimmed.
Importantly, this Umrah is nafl (voluntary) and does not count as the child's obligatory Umrah. Once the child reaches puberty (bulugh), they must perform Umrah again if it is considered obligatory upon them (the Shafi'i and Hanbali schools consider Umrah obligatory once in a lifetime, similar to Hajj). Practical tips: choose less crowded times for Umrah with children, use the upper floors for Tawaf and Sa'i, bring snacks and water, and take breaks as needed — there is no time limit for completing Umrah.