You can create custom polygon shapes from shapefiles using this method here: http://kb.tableau.com/articles/knowledgebase/tableau-polygons-arcgis-shapefiles
A cursory Google search found a couple possible resources for you.
For manual custom polygons, you could using the Drawing Tool for Tableau. Here's an example: https://www.interworks.com/blog/cshelton/2014/10/01/custom-polygons-made-easy-drawing-tool-tableau
Hope that helps!
Matthew, I don't want you to think I'm ignoring you or don't appreciate your response. I'm not and I do - very much so. I was excited to have someone answer me almost immediately with a response that addressed exactly what I'm trying to do.
I've been trying to follow your instructions for the last few days. And have learned a lot in the process. I downloaded a trial version of ArcGIS and have imported the districts of Mozambique file that you found as a layer, which is a filetype ending in lpk. It's gorgeous and exactly what I need.
I have not, however, been able to successfully install the ET Geowizards tool to export the vertices and create a file readable by Tableau. It may be a versioning problem. ET Geowizards doesn't list the current version of ArcGIS as a compatible program. Or it may be just my own lack of familiarity with ArcGIS.
I've tried to export lat-long coordinates for the 100-some districts. The only way I can see to do that is one segment at a time, and each district may have 100 segments. That's a non-starter.
So now I'm flailing around under a deadline and trying things at random. Wish me luck. Thanks for your response. Any further advice you could offer, I'd appreciate.