When the dashboard loads the data, it's just sending a GET to some URL and getting the response, then the dashboard parses the response and renders it. That means the service is sending back data in some format.
If you had the service URL, there's no reason you couldn't alter the GET request and have the response save to a spatially-enabled file like a geoJSON.
It's not a nice built-in feature like the download button, but it is possible to download data from registered services.
- Josh Carlson
Kendall County GIS