Can a woman have a period but not ovulate?

Yes: There is a big difference between the hormone induced changes in the uterine lining that induce a period and the mechanical act of releasing an egg, ovulation.

Not naturally: With medications yes, but not naturally. A true menstrual period is by definition an event that follows ovulation. There are many reasons for vaginal bleeding due to other causes, which can cause confusion, but these are not menstrual periods. It is a safe rule of thumb anyway, that a woman with regular cyclic bleeding is ovulating.