Kissler & Co. Ltd. is a plumbing parts supplier that specializes in parts for older faucets.
You can contact them by phone at 1-800-kissler.
Maybe download their current parts catalogue and look through the drawings of the cartridges they carry to see if you can find a match. If you come up empty, e-mail that photo to them and hopefully someone there will be able to tell you who made that faucet and whether parts are still available for it.
Kissler has a minimum order of $250, but their accounting or customer service departments should be able to tell you what hardware store chains or retailers they regularily ship to in your area, and you can just order the parts from that retailer.