Without serious equipment? Don't think so. I suppose you could go find some gears from something or other and fabricate your own carrier, but you're still not going to do it with a cordless drill and angle grinder. If you want one, they can be had from various parts suppliers and donor vehicles, but you probably don't want one. A well-set up live axle kart will kick the s**t outta one with a diff any day.
I put a diff from an old lawn tractor on the kart I built for my kids, and while it really does give it good pavement manners, it's crappy on low traction surfaces...