A woman may perform proxy الحج on behalf of a man, and a man may perform proxy الحج on behalf of a woman. There is no gender restriction on proxy الحج عند the consensus of the four schools. The evidence for this is the hadith of the woman from Khath'am, where a woman asked النبي about performing الحج for her elderly father, and النبي permitted it without any objection to the fact that the proxy was female and the person was male.
الشيخ ابن باز explicitly stated that gender does not matter in proxy الحج. A daughter may perform الحج for her deceased father, a son may perform الحج for his deceased mother, and so on. The conditions for a valid proxy are the same regardless of gender: the proxy must be Muslim, sane, mature, and must have completed their own واجب الحج. The proxy performs the rites with the intention for the specific person, and the الحج is valid and accepted.
الشيخ ابن عثيمين added that this is a well-أُسس ruling with no significant disagreement among scholars. He noted that some people mistakenly believe that only a man can perform الحج for a man and a woman for a woman, but this has no basis in the السنة or scholarly consensus. The hadith of the woman from Khath'am is explicit evidence that cross-gender proxy الحج is valid.