What type of bolt has a shoulder portion above and unthreaded region below threads?
You are not going to find this on a shelf, you may get one from the manufacturer of the chair, if you talk nice and beg. Otherwise the chance is slim.

I would get a pan or philister head machine screw with the threaded portion about 1" longer than the screw pictured is over all. Chuck 3/4" of the threaded end in a drill and crank it down. Turn on the drill, keep it steady and run a file to grind off the threads exposed beyond the drill chuck to smooth the shank like the one in the picture. Since the screw is longer than needed and the drill damaged the end of the threads to hold it while it was spinning, cut it off and soften the corners like the original. 

The drawing I referred to earlier