Hello Juneoh - A map in a story map must be a web map, either published from ArcGIS Desktop or created in the story using express maps. If you don't have the ability to publish your map as a web map, then a last resort option might be to use a screen capture and just use the image in your story. There really isn't a "migration" rather it's a publish-as-web-map task you face.
Assuming you are a member of an ArcGIS organization, another option might be to go back to the source data and publish the layers in ArcGIS Online/Enterprise and build your web map that way.
Rather than using ArcMap, you should be using ArcGIS Pro - it's the next gen modern desktop GIS and has been designed to replace ArcMap.
