Can you have an oopherectomy at age 15?

YES. Depending on the type of ovarian cyst, if it is medically necessary to have it removed it can be done. What is usually recommended , because of your age, is maximum preservation of viable ovarian tissue.
Oophorectomy . Rarely, an oophorectomy or salpingo-oophorectomy may be performed in a 15 yo because of a benign or malignant (cancerous) mass.