Cane weaving question  

I don't know, as I have never had one of Wegner's folding chairs to dissect. But from having woven cane, I would guess that you could simply wrap the beginning of a new weaver strand around the side post, then run the old weaver strand over top of it, and cut old weaver strand where it passed underneath a warp strand.
Once cane dries it tightens and dries hard in the over-under pattern of the weave, so it does not just slip loose like danish cord would. Thus there is really no need to design a provision for hidden strand joinery like with the CH-25

It might also be possible to put a loose knot where the weavers pass behind the warp. When the cane dries, it would dry hard and not open up. It may be just this simple.

I haven't done one either but I've woven other chairs in binder cane. The usual way to start a new strand is to overlap it with the old one for a few overs-and-unders and make sure the end on top is hidden under an over strand. The end will not pop out as long as your weaving is very compact, which is pretty easy to achieve with flat, slippery binder cane (thankfully!). If you don't pack the strands tightly, they will have room to slide around and two strands layered together could work apart eventually.

And it might just be that incredibly simple.
Due to how rigid the cane becomes, as long as the weaver can get the cane woven into place while soft and wet, then it will harden and provide all the support needed.