Umrah is valid for the children who have reached the age of discernment and for those who have not reached that age but there are certain Umrah
rules for children who travel to Makkah.
A guardian should form the intention of Umrah for a child below the age of discernment (below the age of 7 years) after dressing him in Ihram clothes and keep him away from everything that is forbidden in the state of Ihram. The guardian should make the intention that the child has entered Ihram. It is wrong to form the intention that the guardian has entered Ihram on behalf of the child.
Similarly, the guardian who is performing Umrah on behalf of a child should also form the intention of Tawaf and saa’ee on child’s behalf. He can carry the child during the two rituals. It is preferable to do Tawaf on his own behalf first and then on behalf of the child. But if he performs just one Tawaf, it is also sufficient for both of them. As far as Saa’ey is concerned, it is acceptable to do one Saa’ey on his own behalf and on behalf of the child that he carries. If the child does something which is forbidden in Ihram unintentionally like covering his head or wearing regular clothes, you do not have to offer a Dam on his behalf. On the contrary, if he does it intentionally, for instance wearing clothes because of being cold or so on, then his guardian will have to offer the fidyah on his behalf.
After the child has completed Tawaf and saa’ee, the guardian should perform Qasr i.e. shaving of head or cutting of the hair. Shaving is preferable because the Prophet (pbuh) offered supplication thrice for those who shaved their heads and once for those who cut their hair. If you are afraid that the child may suffer from cold or any other reason, it is sufficient to cut the hair but the hair must be taken from all parts of the head.
Note: It is recommended to consult and seek an opinion from scholars on doubtful Islamic matters.