What is the ruling on a wife obtaining a driver's license if that requires uncovering her face for the photograph, knowing that she is committed to her hijab and will not uncover her face while driving?

If you find it necessary to uncover the face for photography, there is no harm in that, and it is permissible to the extent of the necessity. Photographic imagery differs in its ruling, but scholars agree on the permissibility of what is necessitated by need, such as for a travel passport and identity card.