The WebGL working group has given input to Vulkan pretty early on in the process, but aside from a couple of broad requests the general message was "Don't dumb anything down on our account."
We're too deep into WebGL 2 to be distracted much with what WebGL Next looks like, but certainly the idea of basing it on next gen APIs has been discussed.
Here's the thing: it will probably be impractical to try and expose a ~1:1 mapping for Vulkan and friends on the web like we did with WebGL and GLES2/3. I'm expecting that anything we do build will require some pretty heavy shimming if you wanted to do WebAssembly builds targeting it with Vulkan/DX12 C++ code.
As such my gut feeling is that instead of trying to mimic any particular next gen API on the web we should probably build something that feels web-native (Vulkan definitely does not!) while still retaining the best ideas from the current crop of APIs. Of course that puts a tremendous amount of spec effort on the WebGL working group, and I'm not sure we're currently equipped to handle that.
But as I said, the current focus is WebGL 2. It's a near-term, very realistic goal that will open up significant new GPU capabilities for developers.
Avoiding that work in favor of focusing on a next gen API would be doing the WebGL community a huge disservice.