Can an unmarried woman offer the udhiyah (sacrifice) on her own behalf?

Offering the udhiyah is a confirmed Sunnah for the one who is able to do so, whether man or woman. That has been discussed previously in question no. 36432. It makes no difference whether the woman is married or not. 

Ibn Hazm (may Allah have mercy on him) said in al-Muhalla (6/37): The udhiyah (may be offered by) the traveller as is also the case for one who is not travelling, and it makes no difference. The same applies to women, because Allah, may He be exalted, says (interpretation of the meaning): “and do good” [al-Hajj 22:77], and offering the udhiyah is a good deed. All the people we have mentioned need to do good deeds and are encouraged to do so. We also quoted the words of the Messenger of Allah (blessings and peace of Allah be upon him) concerning offering the udhiyah, and he did not single out the Bedouin to the exclusion of the town dweller, or the traveller to the exclusion of the resident, or the male to the exclusion of the female. So singling out any of these categories is invalid and is not permissible. End quote. 

If a woman wants to offer an udhiyah on her own behalf or on behalf of the members of her household, then she comes under the same injunctions as men, which means that she should not remove anything from her hair, nails or skin, because of the report narrated by Muslim (1977) from Umm Salamah (may Allah be pleased with her), according to which the Prophet (blessings and peace of Allah be upon him) said: “When you see the new moon of Dhu’l-Hijjah, and one of you wants to offer a sacrifice, let him leave his hair and nails alone.” According to another version: “When the ten (days of Dhu’l-Hijjah) begin, and one of you wants to offer a sacrifice, let nothing touch his hair or skin.” 

And Allah knows best.