Esri has verified (April 30, 2018) that this is a bug (BUG-000113626) for the Query Widget in the AGOL-hosted Web App Builder. The problem started probably in late March or early April.
Until the bug is fixed, there is a work-around so users will be able to download the selected records to a CSV file. (Users can still click on View in Attribute Table and see the selected records, they just won't be able to export from there until the bug is fixed.)
HERE'S WHAT THE PERSON CREATING THE APP USING AGOL'S WEB APP BUILDER NEEDS TO DO:
1. In AGOL, edit the application.
2. Edit the settings for the Query Widget. On the "Options" tab, put a checkmark by "Allow to export results" and click the OK button.
3. Save the changes to the app.
4. This will now make four new choices appear to the user when they click the three dots (ellipsis) on the Results pane after they have selected features on the map. The four choices are:
--Export to CSV file
--Export to feature collection
--Export to GeoJSON
--Save to My Content
5. Tell users to click on Export to CSV file and save the file to their computer.