Hajj and ‘Umrah are valid on the part of children who have reached the age of discernment and those who have not reached that age, according to the majority of scholars; in fact it was narrated that there was consensus on this point. 

With regard to ‘Umrah on the part of a child who has not reached the age of discernment, i.e. a child who is below the age of seven, his guardian should form the intention on his part, after dressing him in ihram garments and keeping him away from things that are forbidden whilst in ihram. He should form the intention that the child has entered ihram. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:

He should enter ihram on his behalf, i.e., form the intention that this child has entered the state of ihram. He should not form the intention that he has entered ihram on behalf of the child, because this is not correct. Rather he should form the intention that this child has entered the state of ihram. If he does that, the child has entered ihram.

End quote from Sharh al-Kaafi by Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him). 

Similarly, he should also form the intention of tawaaf (circumambulation of the Ka ‘bah) and sa‘i (going between as-Safa and al-Marwah) on the child’s behalf. He may carry the child during these two rituals. If he does carry him, then it is preferable for him to do tawaaf on his own behalf and then on behalf of the child, but if he does one tawaaf it is sufficient for both of them, according to the correct scholarly view. 

Shaykh Ibn Baaz (may Allah have mercy on him) said: if the one who is carrying another person intends tawaaf on his own behalf and on behalf of the one whom he is carrying, and if he intends sa‘i on his own behalf and on behalf of the one whom he is carrying, that is acceptable according to the more correct of the two scholarly opinions, because the Prophet (blessings and peace of Allah be upon him) did not instruct the woman who asked him about Hajj of a child to do tawaaf for him on his own. If that had been obligatory, he (blessings and peace of Allah be upon him) would have said so.

End quote from Majmoo‘ Fataawa ash-Shaykh Ibn Baaz, 5/257 

Shaykh Ibn Jibreen was asked about that and he said: As ihram for a child is valid, the guardian is responsible for him. So he should dress him in the ihram garments, form the intention of ihram for him, form the intention of the rituals on his behalf, recite the Talbiyah on his behalf, and hold his hand during tawaaf and sa‘i. If the child is incapable, such as one who is very small or is still breastfeeding, there is nothing wrong with carrying him, and one tawaaf is sufficient for both the one who carries and the one who is carried, according to the correct opinion. If the child does anything that is forbidden whilst in ihram out of ignorance, such as wearing a fitted garment or covering his head, he does not have to offer any fidyah because he did not do it intentionally. If he did do it intentionally, such as needing to wear regular clothes because of being cold and so on, then his guardian should offer the fidyah on his behalf.

End quote from Fataawa Islamiyyah, 2/182 

Once the child has done tawaaf and sa‘i, there remains the issue of shaving the head or cutting the hair. Shaving is preferable, because the Prophet (blessings and peace of Allah be upon him) offered supplication three times for those who shaved their heads and one for those who cut their hair. Narrated by Muslim, 1303. But if he is afraid that the child may be affected by cold and the like, then it is sufficient to cut the hair, but hair must be taken from all parts of the head, as is the view of Malik and Ahmad (may Allah have mercy on them both). 

See also the answer to question no. 36847

And Allah knows best.