I would imagine the details are platform specific. I doubt that anyone is using OpenGL, as WebGPU is intended to bring the benefits of the Vulkan/Metal/DX12 generation of graphics APIs to the web.
On platforms supporting Vulkan, WebGPU implementations are probably compiling WGSL to SPIRV before handing it to the driver, and a DX12 implementation would probably compile it to DXIL. I'm not aware of any intermediate representation on Apple platforms, so it's probably transpiled to MSL there.
On platforms supporting Vulkan, WebGPU implementations are probably compiling WGSL to SPIRV before handing it to the driver, and a DX12 implementation would probably compile it to DXIL. I'm not aware of any intermediate representation on Apple platforms, so it's probably transpiled to MSL there.