Fixing pirelli webbing  

http://www.lauritz.com/en/auction/grete-jalk-1920-2005-rosewood-easy-cha...

Rivets.
And it should be pulled tight. Really tight. Consider how much you weigh pushing down on it.

You can use a webbing stretcher to get the webbing tight though in this chair you'll need a spring clamp or two to hold it taut while you tighten it at the next slot---since it's not anchored to the frame at each slot. There's also a type that has very sharp spikes to pierce and hold the webbing but I don't think they are such a good idea with Pirelli webbing.
A pop rivet tool only costs about $20, or maybe you can borrow one.

Thanks guys!