With a little help from Stack Overflow (http://stackoverflow.com/q/21505074/328193) the APPX file upload is now working.  File uploads seem to be a bit of an edge case for WebAPI, and the irony is that the more I think about it the more I think this functionality may intuitively go into a normal MVC controller anyway.  (It doesn't feel right to be performing CRUD operations on the file itself, but rather on the server-side model resulting from the file.  But we'll see how the use cases play out.)  At the very least, the upload issue is unblocked and I can now start picking apart the contents of the APPX file, which was going to be the fun part anyway.