It sounds like you can use "man-made" materials if you make them your own self. So you should research how to make cordage from natural plant fibers. It's not too hard (given the proper raw ingredients), and with a small amount of practice you should be able to make ugly, but functional, rope or twine. This level of technology is certainly appropriate for the stone age; some people have suggested it be called the "plant age" to better describe the human technology of the time.

Evidence of plant-based rope was founded in the Lascaux Cave in France, dating to roughly 17,000 before present.

There were ropes in prehistoric times made from vine or other plant fiber. You can also cut hide into long strips and it will shrink as it dries (so you don't have to tie it too tight initially).

you could use sinew, try a butcher shop. Alternatively gut, like on tennis rackets of old

wedges like this commonly used in hatchets and hammers would be a non manufactured way to fasten tool heads to handles.

Some cultures would weave primitive ropes of grass.

On PBS Nova "Decoding Neanderthals" they talked about neanderthals using pitch(tar) made from birch bark to affix points to spears as opposed to binding with rope.

During the Paleolithic (Stone Age) rope (cordage) was actually a quite sophisticated technology. Dolni Vestonice is a good example of a Paleolithic site in the Czech Republic with evidence of cordage and netting. Other people have commented that sinew was used to bind things, this is also quite common. Paleolithic cordage was probably made from flax fibers (the same plant used to make linen). Other plants that can be processed to make cordage include: yucca, agave, jute, nettle, hemp, juniper bark, cotton, milkweed, etc... You can look up the work of J.M. Adovasio or Irene Emery for more specific information.