Hello Cole,
We decided to deploy ArcGIS Online user accounts instead of the other options at this time. The ArcGIS Online users can create web maps and drag/drop or add csv address files to the map. You can also set this to use your own geocoder so it does not use any credits. This also provide the ability for them save their maps, share, and edit the points. Overall there are a lot of nice features (as I am sure you are aware).
However, the BIG caveat is it bottle-necked by the number of named users of your organization. We have nominated a few individuals from each department to have this access, while others will simply consume the maps created.
Back to the original topic: I did find a couple solutions. One solution which is a simple html web page you can host which supports drag / drop of csv using javascript. It sends your csv to the server and plots markers on the map using Leaflet.js I am a big fan of Leaflet and would personally like to deploy this option, however, due to requiring some knowledge of HTML, the ArcGIS Online option is a safer bet for our initial roll-out. Leaflet is extendable to add other capabilities but excels at providing a simple map for embedding in web sites. I'll look at work tomorrow to find the link and example.
I did attempt the geoprocessing widget / service but there were issues with it accepting a file upload in the Web App Builder environment - there may still be an solution here but I invested a bit of time without success.
The good news is esri knows about this request and it may be in the works.They just added a geolookup widget which takes a CSV input in the Web App Builder. This requires a lat long coordinate and does not run through your own geocoder - so hopefully sometime soon they will post a new or adjusted widget.
Thanks for your interest. Let us know if you find something else.