Those inside washers are usually made from either brass or nickel silver.

I've measured a few from some of the razors I've disassembled and find that they are commonly about 6mm in diameter (about 1/4") with the inside hole large enough to accommodate the rivet without binding. The usual thickness of these inner washers is something on the order of .08 - .1 millimeters (about .003 - .004 inches).

With sheet metal that thin, you can actually use a common hole punch to cut these out (not a hole punch you plan to use for paper in the future, mind). The only difficulty is to first drill the required center hole and then get it cleanly lined up so that it is in the middle of the hole punch die when you cut them out. A little bit of hammering with a ball peen hammer will then be required to flatten out the cuttings, but the end result is really pretty usable.

If you plan to make lots of these, you could also invest in a proper metal punch from a jeweler's supply house. These will give you a really nice result, but plan on paying a fair amount of cash for something good.