Udhiyah is one of the great rituals of Islam, in which we remember the Unity of Allah, His blessings upon us, and the obedience of our father Ibrahim to his Lord. In this act of sacrifice there is much goodness and blessing, so the Muslim must pay attention to its great importance.
In an attempt to furnish you with the answer to your question, we would like to cite for you the following fatwa issued by the late Saudi scholar Sheikh Ibn Baz, in which he states the following:
With regard to the family of the one who is going to offer the sacrifice, they are not subject to any obligations, and it is not forbidden for them to remove anything from their hair or nails, according to the sounder of the two scholarly views. Rather, the ruling applies only to the one who is going to offer the sacrifice, the one who has bought the sacrificial animal from his own wealth.
Moreover, it is mentioned in Fatawa Al-Lajnah Ad-Da'imah in Saudi Arabia that:
It is recommended by the Sunnah for the one who wants to offer a sacrifice, once the new moon of Dhul-Hijjah appears, not to remove anything from his hair, nails or skin, until he has offered the sacrifice, because of the report narrated by the group except Al-Bukhari, from Umm Salamah (may Allah be pleased with her), that the Messenger of Allah (peace and blessings be upon him) said: “When you see the new moon of Dhul-Hijjah, if any one of you wants to offer a sacrifice, let him leave his hair and nails alone.” A version narrated by Abu Dawud and Muslim says: “Whoever has an animal to slaughter, when the new moon of Dhul-Hijjah appears, let him not remove anything from his hair or nails until he has offered the sacrifice.” This applies whether he is going to slaughter the sacrifice himself or he has appointed someone else to do it; but with regard to those on whose behalf the sacrifice is being offered [i.e., his family etc], that is not recommended for them because there is no report to that effect.
Finally, the late Saudi scholar Sheikh Ibn `Uthaymeen said in his book Al-Sharh al-Mumti` :
For the person on whose behalf the sacrifice is offered, there is no blame on him if he removes anything from his hair or nails. The evidence for that is as follows:
1. This is the apparent meaning of the hadith, which indicates that the restriction applies only to the one who is going to offer the sacrifice. Based on this, the restriction applies only to the head of the household, not to the members of his family, because the Prophet (peace and blessings be upon him) connected the ruling to the one who is going to offer the sacrifice, so what is understood is that this ruling does not apply to those on whose behalf the sacrifice is offered.
2. The Prophet (peace and blessings be upon him) used to offer the sacrifice on behalf of his household and it is not narrated that he said to them, “Do not remove anything from your hair or nails or skin.” If that had been forbidden for them, the Prophet (peace and blessings be upon him) would have told them not to do it. This view is the more correct opinion.
The above quotations are excerpted with some modifications from www.islam-qa.com