They say they're made from vegetable material (for the containers, I think) and wood for the window, but I don't know what that actually means, of course.

I meant the containers are just plant based plastics, with no wood in them. There's a whole bunch of processes to derive plastics from plants:

https://en.wikipedia.org/wiki/Bioplastic

Yes, ultimately I suppose all(?) plastics are from plants, possibly via geological processes. I don't know what the legal definition would be, and the packaging which says "plastic free" might well contravene it, but at least it's (apparently) compostable.