Hi -- Sure, you can do that in a story map. You don't need a server, but you will need to publish your data as a hosted feature service on ArcGIS Online.
Once you have that you'll just need to set up separate layers in your map with each of the subsets of features you want your readers to be able to activate. You can do this by copying your layer as many times as you need and using the Filter option in the layer menu to set up a query to show the features you want in each layer. Learn more about filters here.
Once you have those layers in your web map, you can use Story Actions to set up links in your side panel that toggle the visibility of the layers as needed. Learn more about how to create Story Actions here.
Just one more note, from your screenshot it appears you are using Story Map Series for your story (not Cascade), is that right?
Owen Evans
Lead Product Engineer | StoryMaps
