You don't have to use stainless, but when you say "clear coil", I'm assuming you are talking about vinyl tubing. The vinyl tubing will not work, because it's too much of an insulator, so the cold will not penetrate enough to cool the beer. I just built mine using copper tubing. Copper is a better heat conductor than stainless I believe, and I think it may be cheaper. You will want some sort of metal tubing to allow for a good heat transfer.
Also, Welcome to the forums!