Praise be to Allah.
Firstly:
It is permissible for the one who is offering the udhiyah to share the reward of his sacrifice with whomever he wants of his relatives, both living and dead, because of the hadith that was narrated by Muslim, in which it says: “O Allah, accept on behalf of Muhammad and the family of Muhammad”, and the family of Muhammad includes both living and dead. It is also permissible for him to offer an udhiyah on behalf of the dead separately, or along with those who are living. This has been discussed previously in the answer to questions no. 36596 and 36706.
Secondly:
One udhiyah is sufficient on behalf of a man and the members of his household, including his wife, children and parents, if they all live in one house, because of the report narrated by Muslim (3637) from ‘Aa’ishah (may Allah be pleased with her), that the Messenger of Allah (blessings and peace of Allah be upon him) ordered that a ram with black legs, black belly and black (circles) round the eyes should be brought to him, so that he could sacrifice it. He said to her: “O ‘Aa’ishah, give me the knife,” then he said: “Sharpen it on a stone.” She did that, then he took it, and he took the ram and lay it down on the ground, then he slaughtered it and said: “In the name of Allah; O Allah, accept (this sacrifice) on behalf of Muhammad and the family of Muhammad and the ummah of Muhammad.” Then he sacrificed it.
An-Nawawi (may Allah have mercy on him) said: This report was quoted as evidence by those who regarded it as permissible for a man to offer the udhiyah on his own behalf and on behalf of the members of his household, and to give them a share with him in the reward. This is our view and the view of the majority.
End quote from Sharh Muslim by an-Nawawi
Based on that, what is prescribed for the husband is to intend that his udhiyah is to be on his own behalf and on behalf of the members of his household, as the Prophet (blessings and peace of Allah be upon him) said, and that will be acceptable and sufficient on his own behalf and theirs, and they will have a share with him in the reward. There is no need to offer a separate udhiyah on behalf of his wife.
If he did not intend that his udhiyah should also be on behalf of his family, then they are not required to offer udhiyah themselves, because the udhiyah is waived from the members of the household if the man offers an udhiyah, even though they will not be rewarded for something that they did not do and for which the one who did it did not give them a share in the reward.
Ar-Ramli (may Allah have mercy on him) said, concerning the udhiyah: It is a confirmed communal Sunnah, even if it is in Mina, if the members of the household are numerous, otherwise it is an individual Sunnah. What is meant by it being a communal Sunnah, even though it is Sunnah for each of them, is that it is waived if someone else does it, although the reward is not attained by the one who does not do it, as is also the case with the funeral prayer. The author said in Sharh Muslim that if he gives a share of the reward to someone else, that is valid, and that is also our view. The basic principle for that is the fact that he (blessings and peace of Allah be upon him) sacrificed cows in Mina on behalf of his wives. Narrated by al-Bukhaari and Muslim.
End quote from Nihaayat al-Muhtaaj (8/132)
But if the wife has her own money, and she wants to offer an udhiyah at her own expense, then she may do so. If one of her sons gives her money with which to offer the sacrifice, and she accepts that from him, that is also valid.
For more information, please see the answer to question no. 45544.
And that Allah knows best.
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