EDIT: Try xf86-video-ati from here -> http://murga-linux.com/puppy/viewtopic.php?t=48019
(via Arcade Puppy)
It is VERY VERY possible, but not under Puppy. Switch to a full version of Linux, and do it with a couple of apt-get commands. I recommend Xubuntu, or even Crunch Bang Linux (based on Debian Wheezy).
With Ubuntu, you get the PPA repository, which means typing in 4 commands to get it working, otherwise you can use Debian, which may require you patch the source and compile it yourself.
Regardless of how daunting this sounds, there are AMPLE guides and howtos all over the internet. Pick one that looks easiest to follow, print it out, and follow it verbatim. Job done.
Your Puppy is based on Ubuntu Precise Pangolin, so you should be able to install apt (apt-get) and then use this PPA -- https://launchpad.net/~makson96/+archive/fglrx
Read the first few lines, it supports 12.04, which is Precise!
No reason whatsoever that this will not work.
PS. It looks like this driver will not work in xorg >1.12.x, so if you have 1.13 or higher, it will NEVER work
You can downgrade in single-user mode in Precise. You should be able to get the version from the log files under /var/log/xorg.log or x11.log or whatever. Should be at the very top of the file.
PPS. With patching it should work with modern Xorg and kernels
PPPS. If you are serious about getting working accelerated OpenGL, you WILL NEED to perform these steps.
Celeste and Electronic Super Joy