How can I download a shapefile from ArcGIS online web map? I'm looking for "Download to ArcGIS 10 for Desktop", but there's no such option in the drop down list (as attached)
From version 10.3, you will have option, Open in ArcGIS Desktop, once it is clicked, a package file will be downloaded.
This file cane be opened in ArcGIS desktop.
For more information, you can look at below link.
Use ArcGIS for Desktop with your portal—Portal for ArcGIS (10.4) | ArcGIS for Server
The Open In ArcGIS Desktop will only create Layer files with references to the Map Services used in the Web Map and not give you the option to download to Shapefile. As far as I know you cannot download an AGO Web Map. If you want the data in the Web Map, the map author needs to upload the data as a zipped Shapefile or a Map Package and share it with you to enable download.
In order to open in Desktop, the item needs to be a Feature Layer or another 'layer' type of item
From your screenshot, it looks like you're looking at a Web Map.
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