Yes lack of sleep causes acne indirectly. It is however to be understood that lack of sleep has got more to do with the quality of your sleep rather than hours of sleep. It is to be realized that sleep deprivation leads to some important causes of acne like increase inflammation, increase insulin resistance, and increase in stress besides obesity and depression.
So quality sleep can help bring about a positive outlook that could help get over acne. This would also affect overall health, with good health meaning building of immunity to infections or acne. To sum up sleep deprivation makes acne worse, with quality sleep helping to feel fresh and full of energy and kick off acne.