Where do you put multiple washers when you have a bolt and a threaded hole

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

This is sort of a modification of what @FreeMan said, but what I would suggest is you get bolts with flat washers, where the flat washer is less than 20mm diameter so it will fit through the hole. The head of the bolt will need to be smaller than the flat washer (obviously). You will need six pieces total per mounting bolt, including the bolt itself. Put them together in this order: bolt, flat washer, flat washer, (supported structure), flat washer, lock (or split) washer, non-self locking nut. The bolt itself will need to be long enough to accommodate whatever you are going to mount to the vertical structural beams. It would look something like this when put together on the beam:

(NOTE: This is a quick PowerPoint rendering and is not to scale by any means.)

The idea here is to use the bottom half of the washers to grab the wall support. About 1/3 of the washers would be grabbing the wall support, but this will be more than enough to hold it in place. Of course, you'd want to choose the correct length of bolt so it doesn't protrude very far. The lock washer is there so you can tighten the bolt from the nut end without the need for a wrench on the other end, and so it will stay put once it's tightened. If you find that the bolt still moves without a wrench on the other end, you can put a lock washer between the bolt head and the flat washer, but I've found on blind holes like this, if you just hold the threaded end of the bolt with your fingers and use an open ended wrench to turn the nut, the bolt will tighten down without issues. If this is a permanent fixture, you could also put thread locker on the nut to ensure it stays put.

EDIT NOTE: If you are looking for added holding power, move the bolt to the left or right in the hole. This will provide the side and bottom of the hole for something the bolt/washer/nut can hold onto.