What is the ruling regarding a woman who intended to perform Umrah, but then her menstrual period began after she arrived in Makkah? And is expiation (fidyah) due upon her?

Menstruation does not prevent a woman from entering into a state of ihram, but it does prevent her from circumambulating the Kaaba (tawaf) until she purifies herself. Therefore, your wife must wait until she is pure, then perform tawaf, then sa'i, and then shorten her hair to complete her Umrah. She is not required to offer any expiation (fidyah) for menstruating while in a state of ihram.