Is it permissible for my elderly mother to remove her headscarf to obtain the passport necessary for performing Umrah?

Scholars have differed on the ruling of photographic imaging, oscillating between permissibility and prohibition. However, they have agreed on its permissibility in cases of necessity, such as obtaining it for personal documents. Your mother must uncover her face to take a picture if her 'Umrah or obligatory Hajj is contingent upon it. If the Hajj or 'Umrah is not obligatory, then there is no harm in uncovering the face for photography as long as the purpose is legitimate.