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