Could someone get a stress fracture from standing up too much?

Definitely: If the prolonged standing is on a hard surface and you''re not wearing supportive shoes, then it is possible. It's more likely if the prolonged standing is out of the ordinary for that person.

Unlikely: Stress fracture is a result of repetitive micro-trauma, e.g. running. Standing does not generally cause this type of stress on the bones. If you have a fracture from standing or other routine activities, it may be due to low bone density. Please talk to your doctor about your concerns.