The walking foot is part of the feed mechanism. Most machines built or converted for leather stitching are triple feed machines, with smooth feet. There is an inner foot, outer foot and feed dogs. It helps move the material smoothly and evenly through the machine, and the smooth feet wont mark the top surface of the leather as much. You can sew VERY thin stuff on various machines, but anything over 2-3 oz leather should be sewn on a walking foot machine. Getting into machines is a whole different conversation, as there are a plethora of machines with different capacities and purposes.
A common machine saddle makers and others like is the juki 441 and its clones. Its a very heavy duty machine capable of sewing 3/4+ of firm leather. Overkill for a lot of purposes, but there are several holster makers who also like these machines. On the other end of the spectrum is something like the consew 206. A light to medium duty machine based on the flatbed platform. Excellent machine for light bag construction, wallets etc.
Chuck