I don't believe ArcGIS can handle this. However, did find an interesting article, that should allow you to do it with Tableau and a download(free)
I figured out that you can convert them by using the feature vertices to points, add geometry attributes, and then come up with a draw order field for each individual polygon, then exporting as a .csv. Now that is the part I am stuck on automatiing in model builder. I think I need to use a iterate field values tool...
A bit late to the party, but for those reading this now, we (Safe Software) released a free SHP to TDE converter that does exactly what James DuBois asked; namely upload and convert ArcGIS polygons into Tableau polygons without having to download any software. You can try the converter out at www.safe.com/free-tools/shapefile-to-tableau/ It's only been out for a week, so if you have any feedback, please let us know.
If you want to pull in data from other sources, like a File Geodatabase, or need some transformation, we also recently added write support for Tableau Data Extract (TDE) files as of FME Desktop 2016.1.