TWILL TAPE
When I first learned how to make stretch shades, I ordered a big roll of cotton twill tape from one of the shade supply companies, and all the twill tape I'd ever seen prior to this was also made of cotton. It took even longer before I discovered that the packaged twill tape being sold at fabric stores is now also polyester. While you can no longer find packaged cotton twill tape, if you can find it being sold by the yard, chances are it will be cotton (check local home-decor fabric stores and departments).
BIAS TAPE
I also say in the book that you can use bias tape to bind a frame, but in this case, I do point out that you are only likely to find it at fabric stores in poly/cotton, which is a little stiffer, a bit less co-operative, and not quite as attractive as cotton bias tape. Fortunately, there is still gobs of vintage bias tape around (see below for more). I've also found a source for cotton, bias-cut strips that are available in lots of swell colors and tiny prints. It's possible you can find it locally, but see Embellishment Supplies - Ready Bias for more.
VINTAGE BINDING CONSIDERATIONS
As mentioned, Lawrence Schiff Silk Mills is the inventor of rayon seam binding, and according to the company, has been the only U.S. manufacturer since this binding came on the market in 1918. They also said that other brands are privately-labeled Hug Snug, or cheap, imported knock-offs. Hug Snug was their first product, but they later added other rayon bindings, and their "ribbons" (what they call them) have been purchased in bulk and packaged by many other companies over the years. Chances are, any vintage packaged seam binding (made in the U.S.) is either Hug Snug, or another Schiff "ribbon." Since new and vintage Hug Snug are exactly the same, you get the exact same look with either.
William Wright began selling cotton bias tape from a wagon, and then opened a store in New York City in 1897, after coming up with the idea to make bias tape readily available to home sewers by offering it in pre-cut lengths wrapped around a card. Unfortunately, Wright just closed their American factory and moved their manufacturing to China (and moved their office and warehouse to Nashville). If you're interested in learning more about the history of this once-indispensable sewing notion visit the extensive vintage bias tape collection on Fabrics.net (also on Other Sources list).
Vintage tapes and bindings shouldn't be used, if the package is in excellent condition and has good graphics. However, if the package is damaged or non-existent (frequently the case), or the graphics are really ordinary (see Fabric.net for examples of the tape being collected), there's no reason why these materials can't be used. Then, again, if you start collecting this stuff up by the pile like I have, you'll likely end up with lots of duplicate packages, in which case you may want to collect a few perfect examples of each, and then (in my opinion) use the rest. While there are serious collectors who would heartily disagree with me, I think it's better to find a good use for it, then to store it away until it's either sold at your estate sale by the boxful, or ends up tossed in the trash as just worthless old sewing notions.