The washer I originally used was made from plastic. It's also an accessory for arcade stick mechanisms so unless you're into building your own customized joysticks, you wouldn't know how to acquire one. I then tried to see if there was a more commonly available alternative than a joystick washer. That and metal is more durable than plastic, so I wanted to replace the plastic washer with a metal one.
What's necessary for this specific project was a washer with a hole diameter less than 9mm (the diameter of the brass pipe) but larger than 7mm (the diameter of the shaft attached to the Volks chair). Basically, I want the washer to pass through the shaft on the Volks chair but the hole needs to be small enough to not allow the brass pipe to pass. The logic being, just like with the plastic washer, the washer will be flush with the chair's bottom but the hole is small enough to prevent the brass pipe from digging into the chair.
Going to a local home improvement center, I found brass 5/16" washers in a bag. I don't have a caliper but I would guess the hole size is 8.7mm. Since the washer is made of metal, can't be bent out of shape easily, and the hole size can't be increased easily, that size will do. After tax, the bag of washers was 1.29 USD.
Not only is this washer stronger than what I originally used, it's far more aesthetically pleasing and doesn't look out of place.