Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
The family of the one who is going to offer the sacrifice are not subject to any restriction. So, it is not forbidden for them to remove any of their hair or nails.
In response to your question, the late Saudi scholar Sheikh Ibn Baz, states:
Udhiyah is one of the great Islamic rituals. Through udhiyah, we commemorate the genuine obedience of our forefather Ibrahim to his Lord. In this act of sacrifice, there is much goodness and blessing. Hence a Muslim needs to pay attention to its unique significance.
With regard to the family of the one who is going to offer the sacrifice, they are not subject to any restriction in this regard. It is not forbidden for them to remove any of their hair or nails, according to the sounder of the two scholarly views.
Such a ruling applies only to the one who is going to offer the udhiyah and has bought the sacrificial animal from his own wealth.
Moreover, the late Saudi scholar Sheikh Ibn Uthaymeen said in his book al-Sharh al-Mumti`,
There is no blame on the person on whose behalf the sacrifice is offered if he removes any of his hair or nails. The following evidences attest to this conclusion:
1- The apparent meaning of the hadith 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. The Prophet Muhammad (peace be upon him) related the ruling to the one who is going to offer the sacrifice.
Thus, what is understood is that this ruling does not apply to those on whose behalf the sacrifice is offered.
2- The Prophet (PBUH) used to offer the udhiyah sacrifice on behalf of his household. He commanded them not to remove any of their hair or nails. Had this been forbidden for them, the Prophet would have told them not to do it. This view is the more correct opinion.
Allah Almighty knows best.