Probably not that particular brand (Cricut), but the folks who dye discs alot use similar cutting machines. You basically take any Illustrator (or other vector art) file and it will cut it out of sticky-back vinyl for you.
I don't think the "hot-stamping" process works the same way, using vinyl cutouts... but I'm not sure.
You can certainly get a whole lot of expert info in the disc dyeing topic at the dgcoursereview.com forum. That's where I learned how to dye discs.
The big downer in disc dyeing right now is that RIT changed the formula on their dye. The new RIT dye does not work on polycarbonate plastic any more and that has left the do-it-yourself disc dyeing and RC car dyeing community in the toilet for ideas on what to use.
I've reverted to using SpikeIt fish lure dye, but it only comes in a few colors and is pretty expensive by contrast... but if you're just looking to make a disc really visible (hot pink or neon orange), it does a great job.
Thanks for the info Lazer.