Yes, that is permissible for you, because the prohibition on cutting the hair and nails only applies to the one who is offering the sacrifice, i.e. the owner of the animal. The one who is appointed to do that on his behalf is not obliged to adhere to that.
Shaykh Ibn Baaz said:
Those who are appointed to do this on behalf of others are not the ones who are offering the sacrifice, rather the ones who are offering the sacrifice are the one who appointed them to do that.
Fataawa Islamiyyah, 2/316.