Most American straps were leather.
The woven straps were mostly linen. The Canadian/Native style is braided yarn (my grandma taught me this style), and there are a few finger woven and wampum straps known.
The inkle straps made of wool yarn are mostly a reenactorism. They are very easy to make with any kind of inkle loom, tape loom, etc. Set up your pattern of stripes, and make your warp about 1.5 times the finished length. For "dashes" just alternate yarns, like white blue white blue white blue.
This weaving is warp faced, so you have to pull the weft tight so it doesn't show.