If I'm on my period, could I still get pregnant?

Unlikely. I never say never, but typically a normal period indicates that ovulation did occur, but fertilization and implantation did not. Once the lining of the uterus is shedding, it is unlikely to result in pregnancy.
Unlikely but... The fact that you are in menses means the previous ovulation if you had one did not result in pregnancy and the receptive surfaces for implantation are breaking down. That does not mean you will not ovulate during menses with potential to conceive. Women with irregular cycles would have more risk. The womb may not be ready for any implantation. The risk is low but has happened.