Hi Kory
The Story Map apps are really designed to create UIs where the end users aren't provided with that level of control. So we don't expose checkable layer lists because for this sort of UI it is easier if the author creates great looking maps with the best selection of layers for the story being told.
As an author creating a Story Map Series, you can use one or more maps in your story for each entry (tab, bullet, etc) in your series, and you can choose which layers you want to display. So for example you could have different layers turn on and off automatically as people switch between the different entries in your series.
Another option to check out is the Main Stage Actions in the Story Map Journal, This lets you author links or buttons in your narrative that, when your end users press them, perform actions on the map like turning specific layers on or off, zooming to particular extents, panning to a feature and displaying its popup, etc. So that is another way you can allow your end users to interact with your maps: Using Story Map Journal Main Stage Actions | ArcGIS Blog