Is there a list of OpenGL extension support? I am looking for a list that states which vendor does support which OpenGL extension.

For example I wanted to use ARB_shading_language_include, which was accepted in 2013, but from various forums on the internet I assume it isn't implemented on AMD drivers. But I can't find any official document that states where this extension is supported.

Does such a list exist?
There is no official list, but a community driven one. It is largely up to date:
https://opengl.gpuinfo.org/listextensions.php

As can be seen here AMD doesn't support the ARB_shading_language_include extension at the time I wrote this answer.