A woman may perform proxy حج on behalf of a man, and a man may perform proxy حج on behalf of a woman. There is no gender restriction on proxy حج according to the consensus of the four schools. The evidence for this is the hadith of the woman from Khath'am, where a woman asked نبی کریم about performing حج for her elderly father, and نبی کریم permitted it without any objection to the fact that the proxy was female and the person was male.
Sheikh Ibn Baz explicitly stated that gender does not matter in proxy حج. A daughter may perform حج for her deceased father, a son may perform حج 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 حج. The proxy performs the rites with the intention for the specific person, and the حج is valid and accepted.
Sheikh Ibn Uthaymeen added that this is a well-established ruling with no significant disagreement among scholars. He noted that some people mistakenly believe that only a man can perform حج for a man and a woman for a woman, but this has no basis in the Sunnah or scholarly consensus. The hadith of the woman from Khath'am is explicit evidence that cross-gender proxy حج is valid.