I was curious if a bad eardrum can cause vertigo and balance problems?

Highly unlikely: The ear drum is for hearing not balance. An ear drum hole could let water or cold air behind it that could cause vertigo but a hole alone won't affect balance. Vertigo, a perception of you or the environment moving, is usually from the inner ear. If there is a cholesteatoma from a retracted drum it could erode into the balance part of the ear. If you're having vertigo suggest ENT evaluation.