In AGOL Configurable App or AppBuilder, how do I export and download a selected polygon feature in Shapefile format?
It does look like you would need an ArcGIS Server Geoprocessing Service or write a Web AppBuilder Custom Widgets to accomplish this. You may be interested in contacting Esri Support to attach your customer number to this enhancement: ENH-000114908: Adding export shapefile functionality to the Select widget. It's marked as In Product Plan so it does look like we'll see this functionality come into the product.
With ArcGIS Enterprise, you could do the following:
Hope this helps,