What is the ruling on photographing a woman for her passport to travel with her husband for da'wah (inviting to Islam)?

A woman photographing her face for a passport or other purposes is not permissible except out of necessity, because it is awrah (private) and a cause of fitnah (temptation/strife).

Accordingly, if a woman's travel is for the purpose of Da'wah (calling to Islam), it is not among the necessities that permit photography. Women can benefit from male Du'at (callers) abroad without the woman traveling. She should engage in Da'wah within her own environment.

As for a man traveling for Da'wah and his wife accompanying him: if the duration is long, it is permissible for her to accompany him to avoid fitnah. If the duration is short, she should not accompany him. If he fears fitnah for himself, he should not travel and should engage in Da'wah in his own country.