The basemap in Scotland is dislocated around 73m.
Why this happen?
When I open my shape on Google Earth the position is correct, in ArcGis with Basemap is not.
Can you share the shapefile and web map? I suspect that there is an issue with the projection and the basemap aligning as the basemap is in Web Mercator. Are you publishing the shapefile as a feature service?
Technical support can also help look into the specific issue that you are seeing.
All the shapefiles (from my client and from survey data available) have the same problem with dislocation.
This is an example, try use the roads OS OpenData Supply - Download or order OS OpenData and the Basemap provide from Arcgis in the British National Grid Coordinate System. My case is in Scotland.
Also, how can I contact the technical support?
Thanks a lot!
I'm assuming that you are located in the UK, so here is the contact information for Esri UK: http://support.esri.com/contact-tech-support#gb
If I guessed wrong, find the local distributor in your area to contact support.