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