It is low power stuff; 26 awg would work, but that is so small that some may not have the cutting, stripping, and crimping tools to suit.
As far as DC circuits go, oversize wiring reduces resistance and voltage drop, but there may always be some overkill.
For external wiring, the heavier duty wiring makes sense, and there is cloth wiring insulation that looks vintage.
There is sometimes vintage wiring in the vintage lamps or vintage X-mas lights sections in online auction sites.
Fender guitar has some interesting wire for their vintage line.
It has a cloth sheath that can be slid back during the soldering process and then slid back to cover the exposed area when finished.
It also comes pre tinned, which is pretty handy.
The down side, is that it seems to only come in 22awg.
So it is a bit lighter gauge than what was originally being used by the manufacturers back in the day.
Like someone mentioned Rhode Island Wiring Service can provide just about any color/tracer desired in a cloth wire but they cater to the antique auto folks and this is 18 ga. A little thicker but looks and performs well. A little tight for things like Delta/EA handlebar horn buttons but still works and you can match factory schematics (color/tracer). BTW "tracer" is the little 'dashes' seen in a lot of early wiring. Shown below is an example of some wire from R.I.W.S. V/r Shawn
There is such low voltage that the gauge really doesn’t matter. I believe the original wiring was 20 gauge cloth wire but trying to get it through the handlebars And horn buttons is a little tough Especially trying to get them through the truss rods on a silver king. I found some 22 gauge cloth wire on eBay that works fine. Every now and then it comes up.