Firstly: Fulfilling their own Fard Hajj
It is a condition that the appointee have first performed the Islamic obligation of Hajj on his own behalf.  Anyone intending to send someone else to perform Hajj on their behalf must be careful who they select. The appointee should be a religious, trustworthy person such that one is confident that they will full what is wajib. Otherwise, the Hajj counts for him, and does not count for the one who appointed him. This is the position of the Shafi`i and Hanbali schools and the position of a group of the Salaf. This is the chosen position of al-Shinqiti. The Permanent Council issued a fatwa corresponding to this position.
Secondly: Appointing a Woman to Perform Hajj
Appointing others is permissible whether the appointee be a man or woman. This is by agreement of the four schools of jurisprudence.  According to Hanafis, it is makruh for a woman to perform Hajj on behalf of a man. It is the position of the generality of scholars.