How can I export my Web Map as shapefiles?
A web map is usually collection of multiple layers and data sources, including a basemap. So short answer, this is not possible.
You may be able to "print" or save the map as a PDF, or download each layer, if the provider of the data allows this, but the map itself can not be saved a shapefile.
What is it you are trying to accomplish. Maybe "shapefile" is not the correct word or need?
Thanks for the response Rebecca. I'm trying to download maps I made in ArcGIS Online so that they can be edited in ArcGIS Desktop. If you have any suggestions about that, please let me know. Thanks!
That is also not possible. However, you can make a Web AppBuilder (WAB) map online, then download it and modify it some for local web publication. I would wait until the next release or the WAB developer edition if you are wanting to do this, (probably in the next couple weeks) since Online gets updated first, and not all the features are out for the developer edition yet.
If interested in that, learn more about that from links on my blog, which has many links to helpful docs.
Web AppBuilder Developer Edition - Resource List (updated 5/13/2018)
This still may not be what you are looking for, but again, Online to ArcGIS Desktop for editing is not an option.
Adding to Rebecca, you can extract (exportable) layers to shapefiles and other formats (File GDB, KML, CSV). Check Extract Data—ArcGIS Online Help | ArcGIS
Be aware of the service credits that would be consumed.
Click Show Credits before you run your analysis to check how many credits will be consumed.
Moreover, if you are the data owner or admin of the organization, you could also export the data from Item Details page of the item.
Item details—ArcGIS Online Help | ArcGIS
Additionally, you can publish your data as a hosted feature service. If you are using ArcGIS Pro, you edit the data in the web map:
Add a feature service for editing—ArcGIS Pro | ArcGIS Desktop
If you are using ArcMap, you can create a local copy of the data and then sync up the edit to reduce multiple copies.
Making a local copy of a feature service for editing—Help | ArcGIS for Desktop
If you are looking to have others download copies of the data and there isn't a reason to keep the dataset synchronized, use extract data tool or export the data to access the data in a shapefile format as mentioned by Jayanta Poddar
Retrieving data ...