Vertigo is commonly caused by a problem with the way your balance works in the inner ear, although it can also be caused by problems in certain parts of the brain.
Lack of sleep doesn't cause vertigo itself but a lack of sleep can worsen the symptoms of vertigo and make you more dizzy.
Sleep deprivation though can cause lightheadedness and dizziness so it's important to do all you can to get the proper amount of sleep to function properly during the day.