I mainly use the widget to interact with Todoist, but it doesn't show subtasks unless I go to the app. Is there a way I can change this?
Which view are you using with the widget? As it should show whatever is in that view, and sub-tasks do show on all views, even the Today view.
Does removing the widget, and re-adding it resolve it?
I figured out why this is happening, although I don't think this should be the expected outcome. I'm using the next seven days view, and the subtasks don't have due dates (the parent task does). Do I have to add due dates to make them appear?
Yes that would be the case. Date based views need tasks that have due dates,