A browser engine is a compiler. Or, more properly, it's at least two compilers (HTML + CSS -- you can outsource JS to V8 or whatever).
Perhaps you could in an MVP implementation, but in practice no, none of the serious implementations do that by default. First because native OpenGL drivers are generally a mess, so browsers actually implement WebGL on top of DirectX, Vulkan or Metal wherever they can, and even when those aren't available the browsers still parse, validate and reconstitute the GLSL rather than passing it straight through to OpenGL as a layer of insulation against driver bugs. Chrome and Firefox do have hidden feature flags which bypass that behavior and call the native GL implementation directly, but you probably shouldn't enable those unless you're really into ShaderToy and want it to compile faster.
Thank you for the tip!
ANGLE is the de-facto standard library that all of the big browsers use to implement WebGL on top of other graphics APIs, if you want to read up on it.
We do not typically call TeX a "compiler". We don't consider XSLT engines to be "compilers". dot(1) is not normally considered to be a "compiler". And so on.