My question is, is there a pre-compiled Windows binary for Emacs (of the installation wizard flavor) that I could download and get the IT to install for me with a minimum of hassle on his part? I looked at Emacs W32, but as far as I can tell it's Emacs 21, and doesn't have org-mode included.
Get the patched version. It's Emacs 22, and it is the best emacs distribution you'll find on any platform in my opinion.
If you don't mind downloading the org zip file and installing it yourself I think you'll be very happy with EmacsW32. One advantage of installing org-mode yourself is that you can keep it up to date and receive the new features sooner. Btw, you won't need administrator access to install org mode, just EmacsW32 I'm guessing. You also should be able to just run M-x byte-compile-file on the org-mode files to compile them so you won't have to install make.