Users can view the content of a Story map if they are the owner, an editor, or a viewer. If you do not have view access to a Story Map. You will displayed an option to request edit or view access to the Story Map. Selecting one of those options will send a request to the owner of the Story Map, which they can accept or reject.
If a user already has view access they can request edit access by hovering over the view icon in the top left of the Story Map navigation bar and clicking “Request Edit Access“.