Answer:
There are a variety of different ways to make lots of jump rings at a time. One of the least expensive is to use a variety of found items as mandrels, such as knitting needles, old radio antennas, long screwdriver or drill bit, pencil, etc. and an electric drill. Coil the wire around the chosen mandrel shape in the drill, remove the coil from the mandrel and place it into a vise with padded sides (we glued industrial felt inside the vise jaws). Use a jeweler's saw to cut through the rings, lubricating it occasionally (I like bees wax). (We place a towel on the floor under the vise, so when the rings are cut and drop off they don't bounce around.)
If you prefer a machine, then choose the one that will do all of the processes you need, like coiling and cutting. Doing a search for 'jump ring maker', I found a variety of products priced from $70 to $300 and not all of them perform in the same way or do 'everything' you may be looking for, but most products do have both good and not-so-good reviews, which will help you find exactly what you are looking for that fits your budget. (I'll stick with the drill.)
Answer contributed by Dale "Cougar" Armstrong