What's this furniture component called? Metal dowel pin with hole?
If these chairs are from 50 years ago or so the metal pins may very well be 5/16 inch diameter. It turns out that 5/16" is 0.3125" and that is very close to 8mm which is 0.3149". If the peg is a nice fit an 8mm size would not fit into a 5/16" hole. If your pegs are indeed 5/16" in diameter you could try making your own pegs.


Go to the home store and find some 5/16" bolts that have a smooth shank that is of sufficient lentgth to yield the peg length. Such bolts would look like this:




You can cut them with a hack saw without too much effort. A vice to hold the parts can be very helpful. Then use a file to smooth the ends and taper the end corners a bit to get rid of sharp edges. 

A center punch and a hammer can then be used to make a divot into the side of the rod to make a place to start the drill bit to make the cross hole. With a properly located center punched guide it should be relatively easy to drill the hole with an electric drill.

It is possible to also purchase an eye bolt that looks like the following that would also have a smooth shank. With this you would only need to make one cut per bolt and then use the eye for the screw to secure the seat. Since the hole is going to be larger it may be necessary to use a pah head screw and a small washer on the screw.