Limited quantities of replacement parts for vintage appliances are available through specialty stores such as Antique Stoves and Antique Appliances. Online retailers such as eBay are another likely source.
Replacement parts for vintage appliances are tricky to find because they are obsolete. In most cases, even contacting the manufacturers, if they are still in business, does not yield results. The only way to produce replacement parts, then, is to take apart another appliance. Depending on the appliance, the parts supplier needs to be the exact model. Sometimes different models utilize the same parts, though.
Stores specializing in vintage appliances often feature a database of their replacement parts. These companies make their living off of scouring the market for vintage appliances to either refurbish or dismantle for parts. However, such stores also offer reproduction parts constructed to fit a wide range of vintage appliances. If the replacement parts are for the inside of the appliance, such as wiring or piping, then the look is modern. However, for reproductions of exterior parts such as switches and knobs, then the parts resemble the original.
Individual sellers often list their vintage appliance parts on eBay. The stock is not consistent -- it depends on what these sellers happen to have on hand. Customers can search the listings by description or by model number.