Posts on the first page says that the OS on these things is based on Tiny Core. The newer brother MFC printers I had a quick look at on the brother site have "deb" installers for their drivers which might work once converted to Tiny's own format but I've no idea how you'd go about installing them on a machine like this as it's possible all sorts of standard stuff got stripped out because it's aimed at people with no technical knowledge.
If you're doing this to learn linux on a cheap PC you got for nothing then I'd suggest installing a proper modern linux and learn that.
If you're trying to get this working for non tech savvy elderly person to use... I'd probably still change the OS, I can't imagine they're still doing updates for these things 6 years on. If they're not savvy enough to manage a full desktop (linux or windows) then they're probably not going to manage printing and scanning either
If you want to try for the sake of it, here's a forum page talking about converting "deb" files to "tcz":http://forum.tinycorelinux.net/index.php?topic=10844.0