Yes, children of any age can perform Umre, and their Umre is valid as a voluntary (nafl) act of worship. Hz. Peygamber (sallallahu aleyhi ve sellem) confirmed this when a woman presented her child asking about Hac eligibility, and he said: 'Yes, and you will have the reward.' The same principle applies to Umre.
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 Tavaf, walks with them during Say, 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 Tavaf and Say (with assistance if needed), and having their hair trimmed.
Importantly, this Umre is nafl (voluntary) and does not count as the child's farz Umre. Once the child reaches puberty (bulugh), they must perform Umre again if it is considered farz upon them (the Shafi'i and Hanbali schools consider Umre farz once in a lifetime, similar to Hac). Practical tips: choose less crowded times for Umre with children, use the upper floors for Tavaf and Say, bring snacks and water, and take breaks as needed — there is no time limit for completing Umre.