Actually, I've made a small amount of progress since posting my question. When configuring the geoprocessing widget in WAB, under Options there is a check box for 'Use the drawing options of result map service'. If I uncheck this, then the Renderer and Popup options do display for the Output. However, under Renderer, it says 'Can not set renderer because of unknown geometry type.' And under Popup, the list of fields is empty.
The script tool that I am trying to publish is fairly basic, in which I am querying a table, calculating some statistics on the query result, and then joining the stats table to a feature class. At the end of the script I have the line:
arcpy.SetParameter(1, "output_fc")
I checked to see if I overlooked some of the configuration options in the Service Editor when publishing the GP service, and the geometry and schema are blank there also.