No. In ArcGIS StoryMaps, it is not possible to change the story navigation bar in a story map to a collapsible drop-down menu. The story navigation bar is a story element containing links to the main sections of the story map and it is not configurable.
Alternatively, use ArcGIS Experience Builder to create a web app and use the widgets to interact with the data in the web app. Refer to ArcGIS Experience Builder: Widgets overview for more information on the widgets available. For example, the Menu widget is used to organize multiple pages in a web app to navigate from page to page, and the display of the menu is customizable.