It's only possible if the document has been Reader-enabled with LiveCycle Reader Extensions. The necessary usage right cannot be applied with Acrobat.
I don't know anything about LiveCycle. Is what you are saying the same as using Acrobat X Pro File -> Save As -> Reader Extended PDF -> Enable Additional Features?
If so, that part is done. But it seems I need to run the command exportAsXFDF in a privileged context. If I can do that at all in Acrobat Reader. That is what I'm trying to find out.
I don't know if it isn't working because I'm just not doing it right or because it isn't possible in the first place.
No. There are two things that can Reader-enable a document. One is Acrobat using the method you described, and the other is LiveCycle Reader Extensions (which is not the same as LiveCycle Designer that comes with Acrobat Pro for Windows.) LiveCycle Reader Extensions is able to include more usage rights than Acrobat can, and the one you need is one of them. So Acrobat is not sufficient for this. You're correct that the script will need to be executed in a privileged context.
An alternative that does not require any usage rights is to set up the form to submit as XFDF to a web server.