You can mine slate without a pickaxe, so you can easily get stone tools without wood.

What version?

I just tested this, and it happens on 1.16. You can not only mine slate with your bare hands, but also the other Quark stones (limestone, marble, jasper, brimstone, voidstone etc.).

Quark: Quark-r2.2-249
ARL: AutoRegLib-1.6-41
Forge: 32.0.62