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