No, you cannot use Natural Cycles while you are using hormonal birth control.
In general, the hormones in hormonal birth control affect your temperature. As Natural Cycles uses body temperature to calculate your fertility status, the app cannot be used together with any hormones that impact your temperature.
Additionally, hormonal birth control generally works by stopping ovulation, which means that you don’t have an actual menstrual cycle while using hormones. The bleeding you may experience while on hormonal birth control is not considered a period; it is a type of spotting called withdrawal bleeding. If you would like to read more about when and how to start using Natural Cycles after hormonal birth control, we have an article about this here.