Is it permissible for the mother of the wife —who is an elderly woman (qa'id)— to travel by plane without a mahram to take care of her daughter who is in postpartum bleeding (nafsā'), knowing that the travel of a mahram would increase the cost, and it is difficult for the couple to move to their country for the childbirth?

The authentic Sunnah (definition) prohibits a woman from traveling without a mahram (definition). A woman's age does not affect this ruling; the prohibition is absolute, even with trustworthy company. However, some contemporary scholars have permitted a woman to travel under specific conditions and when there is a valid Sharia (definition)-compliant excuse. If there is a genuine need for your mother-in-law to travel without a mahram, while observing the conditions, some scholars have allowed acting upon scholarly dispensations when necessary to alleviate hardship.