Why do my ankles hurt after a vigorous workout?

DIfficult to say: You would need a full ankle exam by a qualified physician to make an educated diagnosis about ankle pain following a workout. See either a podiatrist, a physical medicine specialist, a sports medicine doctor or an orthopedist for exam.

If your pronated: For example you'd be putting extra strain and stress on them. Try an orthotic.

Orthotics may help: It sounds like you could also have a biomechanical instability in your feet that is passing problems 'up the kinetic chain.' custom made prescription orthotics, made by a certified lab, under the recommendations of a podiatrist can help tremendously!