Is it true that if a girl starts having sex early, she will have less chance of certain cancers?

Nice myth. To the extent this may increase the number of partners and related STD's, this may be the opposite. She may actually increase her risk of cervical cancer.
Pregnancy, OCPs. Pregnancy and ocps (oral contraceptive pills) can reduce the risk of ovarian cancer. Risk is reduced by about 10% per pregnancy. If sex is started early, both these things can happen, and so risk for ovarian cancer is reduced, but not by sex itself.
Pregnancy, OCP's. Pregnancy, and use of ocp's (oral contraceptive pills) reduce the risk for ovarian cancer. Risk is reduced by about 10% per pregnancy. I guess if sex is started early, the use of ocp's is started early, and so risk is reduced. Needless to say, pregnancy can also occur if the ocp's are not used. But sex itself does not reduce ovarian cancer risk.