You don’t have to worry about whether it’s open or not—as long as you don’t manually close it by swiping up in the multitasking view. When you do that, iOS won’t allow the app to run in the background until you open it again. In other situations, iOS will automatically run the app in the background if it’s not already open, so it can update and show notifications.
Deliveries uses a feature called “Background App Refresh” to update your deliveries in the background. It runs for just a few seconds every 1 to 3 hours, depending on how soon your deliveries are expected. We’ve worked hard to make it very efficient, so you don’t have to worry about wasting battery life.