It is permissible for the family of the one offering the ‘udhya to trim their nails and cut their hair. As for the one offering the sacrificial animal, it is recommended that he does not cut or remove anything from his hair, nails or skin. There are divergent opinions from scholars regarding the matter. Some prohibited it, some disliked it (as makrooh), while some permitted it. The ruling is solely on the owner of the sacrifice; his family is not included in the ruling.
Ibn Baz said: As for the family of the person offering the sacrifice, nothing is binding on them. They are not stopped or forbidden from shaving their hair or cutting their nails according to the more correct of the two scholarly opinions. Rather, the ruling concerns solely the person offering the udhyah especially the one who bought the animal with his money. (Fatawa Islamiyya 2/316).
It is also available in the fatwa of the Standing Committee for issuing fatwa in Saudi Arabia (11/397)
It is prescribed for the person who wants to offer the udhyah when he sights the moon of Dhul Hijjah not to take anything from his hair, nails or skin until he slaughters this sacrifice. This is based on what was related by the Group of the Compilers of hadith excluding Al-Bukhari (may Allah have mercy on them) from the authority of Ummu Salmah (may Allah be pleased with her) was that the Messenger of Allah (peace be upon him) said: “If anyone has in his possession sacrificial animal to offer as a sacrifice, he should not get his haircut and nails trimmed after he has entered the first 10 days of Dhul Hijjaah.”
Another version from Abu Daud (2791) and Muslim (1977): “when anyone of you intends to sacrifice the animal and enter the month of Dhul Hijjah, he should not get his hair cut or nails trimmed till he has offered his sacrifice.” Whether he slaughtered it himself or he assigned the slaughtering to someone else. As for the person who is making the slaughtering on people’s behalf, the ruling is not prescribed for him because no verdict was received about that.
Ibn Uthaymeen said in Sharh al-Mumti’ (7/530):
For the one whom the sacrifice is made on his behalf, there is nothing wrong with him taking from the above mentioned, and the evidence for this is the following:
1. That is the apparent meaning of the hadith. The prohibition is specifically on the one offering the sacrifice and that is the head of the house (family). As regards the household, nothing is prohibited for them because the Prophet (peace be upon him) emphasized that the ruling is binding on the one offering the sacrifice. So, what is understood from it is that those whom the sacrifice is made on their behalf has nothing to do with the ruling.
2. The Prophet (peace be upon him) used to offer sacrifice on behalf of his household, and he was not reported to have said: “do not cut your hair, nor trim your nails, nor take anything from your skin.” If those were haraam for them, the Prophet (peace be upon him) would have forbidden them from cutting or trimming, and this statement is the most correct view.