Some OEM carries reinstall media for years after the computer was manufactured. While it's highly unlikely they're still producing this media, there's always the chance it's in stock, if all copies weren't sold or requested for warranty support. Most throws nothing away, so if it's there & can be converted into cash, they'd gladly furnish it for a nominal fee.
One thing that I will warn you about. the XP Media Center reinstall is the trickiest of all Windows installs that I've tried. It's odd, being that some of these systems were released before SP2 for XP was, that during the clean install, many will request an XP SP2 CD to be inserted. In other words, one begins the install with CD #1, it then requests CD #2, then wants a XP install CD of SP2 (any type will do) for a few files, then swap back to CD #1 & complete.
However there's a few where everything is on a single DVD & hopefully in your case, all of this swapping won't be needed. The first time I did this, was confused by the request for SP2 media, didn't really know what to do, so I inserted a Dell OEM XP Pro SP2 reinstall CD (it was all that I had), the few needed files were fetched & it worked afterwards.
The XP Media Center Edition in essence, is a modified version of XP Pro. This was a pricey PC at the time of release, the couple of people that I know whom has it, ended up upgrading the video card & RAM, then installed Windows 7 Home Premium, which includes Windows Media Center out of the box.
So if JohnC_21's advise gets you nowhere. contacting Gateway is an option. There is also the chance that by entering you model number & finding your computer, where the drivers & optional downloads are, there could be a "Recovery Media" link on the page. That's how it is with HP. Dell also has such links. If you make it this far, your serial number will be required, so be prepared to provide it & other necessary information asked for.
If you can't get anywhere with what's provided, then we'll need to explore other options.