I love natural slings, I've never actually used synthetic materials except for once early in my slinging days when I made a leather pouch slings with paracord for retention and release, and that didn't turn out so well. I used to have a problem with the end of the release cord coming undone due to durability of the material, though I've started using horse hair for the end and that's proving to be a lot more durable than other fibres.
I don't know which country you're from, but here in Australia it can be pretty tricky to find high-quality natural materials. There are some online suppliers I've found of hemp and linen fibre, and sisal string (though i had to do some digging). I've never been to a store that's sold anything but jute twine and cotton thread, so all my shopping for good material has to be done online.
Here's a site that was linked to me in another thread
https://www.artandcrafty.com/en/They sell esparto and by the looks of it plenty of other natural fibre, and pretty sure they can ship to most of Europe if you're there. Good luck!