How do I fix/fasten pirelli webbing?  

That's my "before" photo of a chair that I redid a few years ago. The original webbing was extruded vinyl (photo below is of one that sold online, the only one I could find that showed the original webbing).
Do the slots accomodate the width of Pirelli webbing? I honestly don't remember. I do remember hunting around for something that would fit the slots.
You might be able to do anchor the webbing in each end by sticking a loop of it through the slot and inserting a dowel through the loop---IF the slot is wide enough to fit two layers of webbing through it. I don't remember this, either, but it is easy enough to check.

Looking around it seems a lot are done like this:
http://www.lauritz.com/Item/ItemImage/ItemImagePage.aspx?LanguageId=4&It...
http://www.lauritz.com/Item/ItemImage/ItemImagePage.aspx?LanguageId=2&It...

I just removed the cheap nylon straps from the Johannes Andersen P Jeppesen lounge chairs that I picked up a month ago. The seat support has a very similar design to the Jalk lounge chairs. I can confirm that the slots on the front are offset by half a module from the slots on the back (4 in front, 5 in back). Therefore, I think the zig-zag "W" pattern for a single strap is probably the correct original pattern.
On the underside, there is no groove below the slot where a cinch strap tightening (copyright to leif) would go. Mine had a DIY clamp by wrapping the strap around a scrap metal strip, then screwing it directly to the wood, so I cannot help on original strap fastening design.

The zig-zag pattern is definitely original and also very unusual.
I really like the riveted ends clearly visible in the 2nd photo link by Leif. Or maybe they're not rivets--?? Anyway, you can buy a Pop Rivet tool for under $30, I think. That should work nicely. Or maybe use a few threaded connectors----not sure that that's the right name? Threaded post connectors? One looks like a screw or bolt but the shank is hollow and threaded inside, and the other is a screw that screws into this.

There are a fair number done with those rivets. Enough that I suspect that is the original technique.
Oddly they are seem to be done in different materials, and many of them seem very old. I wonder if the extruded vinyl was not aways the material used?