Vintage woven "string" chair restoration  

I am at wit's end - I've searched for two years attempting to find a source for material to "restring" my collection of vintage patio furniture. This furniture is steel, and predates the cheap "vinyl strap" furniture seen everywhere.

The cording I need looks like the material used on these chairs:

This cord (also used on recent IKEA chairs as well as other modern furniture) needs to be flexible but not elastic, 1/8" to 3/16" diameter, good colors, and available in retail quantities. I could also use 1/4" straps of a similar material. I don't think this material is the same as what's currently sold as "resin wicker", which seems to be stiffer and less flexible - though that may work as well.

Please - any help here would be very much appreciated.