Very simple
for my project I used 2 lengths of rawhide and a small patch of leather.
one length maybe half your height or so, and the other should be a good 8 inches or so longer.
punch holes in the patch of leather, push the raw hide through and knot the ends so it wont come out.
now take the longer of the two straps and wrap it around your hand, hold the other strap between your thumb and fore finger. put the object to be thrown in the strap and use centrifigual force to keep in in place as you spin it. when it is time to release let go of the shorter strap. The longer strap around the hand will keep you from throwing the sling.
thats pretty much it. Im sure you can improvize the materials from your kit if need be. If you are in some place with a lack of rocks like mississippi then things like batteries make good ammo. The spin up and release is similar to the boleadoras mentioned in another topic, you should with practice be able to do it in a quick fluid motion.
once again my materials were salvaged. One good source for nice leather are saddle companies, check for them in your area and inspect their dumpsters, you can find scraps that are very useful for sheaths and whatever else you can imagine. (ot but you can use leather for gaskets on your vehicle if need be)