This is a great question. Thank you for asking.
I think you could do this by setting an alert so that every with the "as frequently as possible" setting on a condition that will always be true then you will get the email each time a refresh completes. I am testing this but I think that should work.
I am moving this to Server Administration to see others ideas.
A different solution might be to do what I do for the server that I manage, I use a local pc to do scheduled refreshes of the extracts with 'tableau -refreshextract' option and then I pipe the output to a text file. The text file is sent by email to an alias. The output will show the number of records in the new extract and if it is blank then this indicates the refresh failed. It's very handy.
tableau refreshextract --datasource "nameofdatasource" --site "SiteName" -c RefreshConfig.txt >> text.log
I use mailsend to send the log file.
I'd love to hear what others are doing as well to build up some more shared knowledge here.
I recommend VizAlerts:
Get alerts "when the extract refreshes"
If you've wanted to run your VizAlerts when the extracts in the workbook they're a part of refresh, you've got your wish. The administrator may now set up two schedules; one for Refresh Success and one for Refresh Failure. Subscribe to either or both of those, and your VizAlert will run when the extract(s) in the workbook refresh."
Me and a couple of my end users take advantage of this nifty tool.
Thank you all for the responses.
I was able to get access to a workbook that mimics the "Background Tasks for Extracts", and adjusted it to create a report that provides the data I needed for this.
Thank you for sharing with the community what works for you and marking the post resolved/answered.
Have a great weekend.