How can I secure stiff fabric to wood without risking ripping and without adhesives?
Instead of securing the fabric with the scews, I'd let the fabric secure itself on the outer edge of the wood. The less holes you poke into a fabric, the stronger it is to withstand tearing or wearing out.

Instead of a simple hem, you could roll and secure the fabric (light blue) around a cord or other thick material (dark blue) to create a bulge that is held by the wood (brown). If the material is somewhat thick and the wood closes well without big gaps, a "simple" hem might even be enough.



Advantages:


Preserves the fabric's strength by avoiding holes
If a weight pulls the fabric down, the force is distributed between all points on the edge instead of single holes. 
You can easily replace one piece of fabric with another one.
No need to sew button holes or inser grommets into the new fabric


Disadvantages:


You need to sew a thick or corded hem on the fabric
The hem is always visible on the "outside" of the wood pieces