Hi! Nope, shaving (or depilating/waxing/any other form of hair removal) your vagina would not cause you to develop an ovarian cyst. It does however put you at risk for some skin conditions, which can be brought about by irritation or a reaction to the products you apply on your pubic area.

Ovarian cysts are usually due to hormonal imbalances (may manifest as irregular menses), obesity, and other gyne conditions.