You have a few ways to get custom symbols.
1) If you have an organizational account you can upload feature service through ArcGIS Desktop, with the symbols you want.
2) If you have ArcGIS server you can also host a feature service and just import it to your map. No organizational account needed.
3) The last way is to upload your zipped shapefile and then use a custom symbol instead of the once offered by ESRI. All you need to do is upload your symbol somewhere and then point to it in the custom symbol option.