Mint’s default wallpaper manager doesn’t, and Discord doesn’t let me pick a .webp as an avatar. Those seem like 2 pretty big ones that don’t work.
I’ve also run into other less common examples over time, but those are more random spread out things and I don’t remember what they are.
I can’t speak for all distros and DEs, and I also don’t do many image related things, but I’m using Linux Mint Cinnamon and the default desktop background manager doesn’t support .webp. Sometimes I see a cool image that I want to use and I have to convert it; other times, when I notice it’s .webp, I just give up on that image.