I purchase the app on the App Store. Do I have to pay for a membership?

No, you don't. We went to a subscription model for a short time last November 2019, because we thought it would be a way to help us keep improving the app. However, our users let us know that they did not like this model, so we decided to go back to the previous model. It seems we were wrong, and so Story Planner is a one-time price app again.