I attached a USB drive large enough for the Mountain Lion installer (8GB) to my 2012 MacBook Air that shipped with Mountain Lion.
Short answer is that nobody knows how to do it at Apple Support or the App Store.
I was finally sent to the App Store support who then sent me back to Apple Support. I've been on the phone with Apple Support for almost two hours explaining this and why I would want an installer for the OS my computer shipped with.