For heavier items like bags and slings, woven linen strapping can be found. I think Susan Wallace at Silver Shuttle Memphis TN has this.
It's not as tightly woven as the old linen was, but still very serviceable. Comes in different widths.
Likely Jas Townsend and Samson Historical will have similar.
For lighter objects, leather boot laces if oiled and whipped around a post to soften and round them, works very well, or strips cut from veg. tanned leather and greased with deer tallow to take the new off them , work well.
This is the I think 2 inch. doesn't come with the happy little flowers!