She does not have the right to allow her face to be photographed, whether in a passport or otherwise, because it is ‘awrah, and because having her picture in a passport or elsewhere is a cause of people being tempted because of her.
But if she cannot travel for Hajj without doing that, then she is granted a concession allowing her to have her picture taken so that she may perform the obligatory duty of Hajj, and it is not permissible for her to delegate someone else to do it on her behalf.
The entire woman is ‘awrah according to the evidence of the Qur’an and Sunnah, so what she is required to do is to cover all of her body before non-mahrams, because Allah says (interpretation of the meaning):
“and not to reveal their adornment except to their husbands, or their fathers, or their husband’s fathers …”
“And when you ask (his wives) for anything you want, ask them from behind a screen, that is purer for your hearts and for their hearts”
Shaykh ‘Abd al-‘Azeez ibn Baaz,
Shaykh ‘Abd-Allaah ibn Ghadyaan,
Shaykh ‘Abd-Allaah ibn Qa’ood.
Fataawa al-Lajnah al-Daa’imah (1/718, 719)