Praise be to Allah.
Hajj is obligatory upon the adult, on condition that he or she is able to do it, whether that adult is a man or a woman, because Allah, may He be exalted, says (interpretation of the meaning):
“And Hajj (pilgrimage to Makkah) to the House (Ka‘bah) is a duty that mankind owes to Allah, those who can afford the expenses (for one’s conveyance, provision and residence)” [Aal ‘Imraan 3:97].
The verse is general in meaning and indicates that Hajj is obligatory upon the one who is able to do it, whether that person is a man or a woman.
An- Nawawi (may Allah have mercy on him) said: The scholars are unanimously agreed that Hajj is obligatory upon a woman if she is able to do it.
End quote from Sharh Muslim by an-Nawawi (4/148)
Based on that:
There is no specific age for Hajj. If a person has reached the age of accountability and is able to do Hajj in financial and physical terms, then he must do Hajj immediately. For more information, please see the answer to question no. 41702.
In addition to the conditions of being able to do Hajj, in the case of a woman it is stipulated that she should have a mahram with her. If a woman cannot find a mahram to do Hajj with her, then in that case she is not obliged to do Hajj, even if she is financially and physically able. For more information, please see the answer to question no. 83762.
A child is not required to ask his or her father for money in order to perform Hajj, but if the father chooses to give him or her money for Hajj, there is nothing wrong with accepting it and doing Hajj with that money. Similarly, if the father makes a booking for one of his children in a Hajj group, and pays for those expenses, then this is something good and comes under the heading of spending that is prescribed in Islam, and there is nothing wrong with the child accepting such an offer, especially if the child did not ask for it.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked:
I am a student who has reached adulthood, and I do not have any wealth of my own. Can I ask my father for money in order to do Hajj now, or should I wait until I graduate and find work, so that I can do Hajj with my own money, even though that will take a long time?
He (may Allah have mercy on him) replied: Hajj is not obligatory for a person if he does not have any wealth, even if his father is rich, and he does not have to ask his father to give him money so that he can do Hajj. Rather the scholars say: Even if your father gives you money with which to do Hajj, you are not obliged to accept it; you can refuse and say: I do not want to do Hajj yet, and Hajj is not obligatory upon me (at present).
Some of the scholars say: If someone – a father or a full brother – gives you wealth with which to do Hajj, then you must accept it and do Hajj with it.
But if someone else gives you the money, and you fear that one day he will remind you of his favour, then in that case you do not have to accept it and do Hajj with it. This view is the correct one."(Majmoo‘ Fatawa Ibn ‘Uthaymeen 21/94).
For more information, please see the answer to question no. 3463.
With regard to appointing someone else to do Hajj on one’s behalf, this is only applicable in the case of one who is physically unable to do Hajj because of old age or sickness from which there is no hope of recovery. In this case what is prescribed is to appoint someone to do Hajj on one’s behalf, if one has wealth.
But in the case of one who is physically able but does not have wealth, or if a woman is financially and physically able to do Hajj, but she does not have a mahram who can do Hajj with her, then in this case appointing someone to do it on her behalf is not valid, because Hajj is not obligatory in the first place in these two cases, and because a woman may be able to find someone among her mahrams who can do Hajj with her in the future.
It says in Majmoo‘ Fatawa Ibn Baz (16/122):
It is not valid to appoint someone to do Hajj on behalf of someone who is physically healthy, even if he is poor, whether that Hajj is obligatory or supererogatory. As for the one who is unable to do Hajj because of old age or sickness from which there is no hope of recovery, then he must appoint someone to do the obligatory Hajj and obligatory ‘umrah on his behalf, if he can afford to do so, because of the general meaning of the verse in which Allah, may He be glorified, says (interpretation of the meaning):
“And Hajj (pilgrimage to Makkah) to the House (Ka‘bah) is a duty that mankind owes to Allah, those who can afford the expenses (for one’s conveyance, provision and residence)”[Aal ‘Imraan 3:97].
For more information on the rulings and guidelines on appointing someone to perform Hajj on one’s behalf, please see the answer to question 111794.
And Allah knows best.