Can you customize a tour Story Map to have an action, like turning on a layer?
Story Map Tour doesn't support creating Story Actions, like Story Map Journal does (see blog post about Story Actions in Map Journal).
You can include supporting layers in your Map Tour by adding them to the web map used in the Map Tour (you make that edit directly to the web map via ArcGIS My Contents, not in the Story Map Tour Builder). Supporting layers are displayed with their scale visibility rule, if that has been defined in the web map, so they'll only appear at particular scales in your Map Tour. So that is a way you could have a layer appear as people proceed through your tour. But there's no way to configure the Map Tour to allow users to manually turn a layer on. I had a look in the Story Maps Gallery using the Format > Customized filter to see if I could find a customization of Map Tour where people had add done development to that capability but I couldn't find one.
People have added a Basemap Picker to the Map Tour via customization, such as: http://maps.simcoe.ca/Trails/?TRAIL=MatchedashBayLoopTrail
Retrieving data ...