Replacing webbing on Folke Ohlsson lounge chair  

Can you remove one of the torn straps and take a photo? These Fagas straps should have a 45 degree metal strip which locks into a corresponding groove in the wood seat frame. Usually the groove is continuous, but unless there is a different connection for these Dux chairs, the groove looks like it might be intermittent.

If you want to replace with similar Fagas straps, you can order here: http://www.fagasstraps.com/

Otherwise, Pirelli or Elasabelt webbing would work just fine as well.However depending upon the width of that groove in the wood, the standard clips for these webbings might be too thick for the groove.

Just added the photo! It's a 45 degree angle. There is one groove per strap, not continuous.

Also, the size of the part that enters the wood is about 1/4" and the width is 2".

This is the measurement of your clip that is crucial. The ones made by Osborne that you attach yourself are most likely going to be too thick to fit into the groove in the wood. (I made that mistake once.)