Quoted from Chet:
BAA sells stranded wire in the correct gauge and color/trace that will replace the original wires in williams games. They sell in std 10 feet lengths but you can also request longer if you need it. I have used there wire to rebuild transformer harnesses, etc., to duplicate the original.
BAA didn't have the wire colors I was looking for, so I hunted around for alternatives that didn't require a 1000ft minimum order. That took a considerable amount of searching and communication with various companies.
One of the few places I found was narragansettreproductions.com---it's an antique car restoration shop in Rhode Island that specializes in wiring harnesses.
They have a minimum order of 25ft per colored wire for 16awg ($8.75) and 18awg ($9?) wire. Also, a 250ft minimum order per colored wire for 22 awg wire ($26). They don't give a shipping estimate until it is sent out, but they sent my order in a standard USPS flat-rate box (which are currently $5.95, $12.65, and $17.90 for small, medium, and large).
They typically have 16-18awg wire in stock, but will likely have to special order 22awg wire since they don't keep it in stock (since it's not really used on antique cars), so this is the reason for the 250ft minimum order.
Also, their ordering system is a bit old school...they take credit card info by phone, and I'm not sure, but they might take mailed checks too--you'll have to ask if they do that.