here's some cheap PC ways of doing it. First, you can use kite string to sew bags. I use it for everything because its 100% cotton, cheap, and easy to find.
Another thing, use elmer's glue to tack down the fabric if you don't want to go with a running stich. They had horses and cows and the rest of the ingredients for the glue back then also- its really only a cleaner version of hide glue so it will be ok. If people don't think that'll past muster, then use it sparingly(you should anyways because sewing through that glue is a pain in the keister) and when you're done sewing it, give it a rinse. Elmer's glue is water soluble so it will wash right out.
Lastly, I don't claim to know much about white-made hunting bags, but I know that indian pouches (when lined) used an array of fabrics, some even being unmatched scraps that were pieced together. If you have any period correct fabric that might be a route to take. Unless you're going for a bag that was made by a company of some sort.
Hope this helps and as with all stuff, take it with salt.
Scouter