Look for a link near the top of the page which says "click here to download." This is the manual download link, intended for when the automatic installation process fails. Click the link to bring up the download window. It will prompt you to download the file to your local hard drive.
Use a flash drive to copy all the Adobe Reader files to the non-internet-enabled machine. This also assumes that none of these updates are registry-related, but these steps are relatively easy to implement and the worst case scenario is you'd have to reinstall Adobe Reader.
We did that. The first person we were chatting with said he was the wrong person and transferred us. The second person ALSO transferred us. I didn't even realize you could transfer a chat session. All-in-all, the third support technician figured out a way to install the software without using an Adobe ID.b2IN1FB.