Well ok, two ways to go here. First is the classic, all metal machines. Flush mounted tables, external motors. Getting kind of old these days, but believe it, they will outlast us both. They are relatively cheap, replacement parts ridiculously cheap, but it helps to be a bit of a tinker-er to really keep them up and running. The Singer 31-15 is ok, but no reverse. I'd look at the equivalent machine in the next gen, such as Mitsubishi, Consew, etc. The biggie is getting a machine that takes a min of a # 18 needle (with 20-22 being optimum), at least 3/8" presser foot height, a LOW speed motor (under 1100 rpm preferred), and the smallest pulley you can fit on the motor. Uses #69 or size "E" thread.
Other guys step it up to a "walking foot" or double-feed model, which is good for heavy build-ups and the stupid things we do. Preferred for production sewing. But for home custom work I still like a simple single feed. These can go up to "3 cord" for seriously heavy work. There are even heavier machines, but we'll stop right here.
Now the other way of going is "table-top" or internal motor designs. These are basically your home sewing machines, usually metal/plastic blends. Also "Sail Rite" table machines, made for light weight canvas repair. If you watch what you're doing, these will work, however it pays to stay on the heavy-duty side of these. Singer has a nifty little HD machine you see at places like Jo-Ann's and so forth. As long as you can get a #18 needle, and "heavy-duty" nylon thread through it (almost #69), you should be able to make some basic things.
Now believe or not, sewing is sewing, so start talking to home clothes-makers, quilters, upholstery and drape makers, anybody you can find that sews. They all have T,T,P's you can use. Get the latest FAA Parachute Rigger's Guide by Sandy Reid (Vietnam Recon Marine BTW). And/or Poynter's Parachute Manual, Vol 1. Solid info on machines, materials, and techniques directly applicable to nylon gear.