No. Recent recommendations from the American College of Obstetricians and Gynecologists state that teens and young women do not need annual Pap tests, regardless of whether they are sexually active.
Girls under the age of 21 are at extremely low risk for cervical cancer, which is the primary reason for getting a Pap test. And in fact, early screenings may do more harm than good, because they can create false positives and lead to unnecessary follow-up procedures at a rate astronomically higher than the actual risk of getting cervical cancer. So following the medical creed “First do no harm,” having teen girls undergo the test is counterproductive.