Welcome to the forums @phonetext!
You are building from master, correct?
With master, you will need to have Vulkan drivers installed on your computer, if you do not already. AS far as the error, it looks like it has something to do with a standard library file (stddef), so I would ugess that it is something to do with the C++ compiler you are using. I would double check the compiling instructions on the Godot documentation and see if there is anything there you have missed.
For Android, I do not think there is support for Godot 4.0 on Android just yet. While Android does have Vulkan support, I believe it has not been implemented as of right now. OpenGLES 2 and/or OpenGLES 3 support will also be part of Godot 4.0, but neither has been implemented just yet. I think Godot 4.0 isn't quite ready for Android yet.
That said, I could be completely wrong about this! I have not tried to compile for Android on with the Godot 4.0, nor have I been keeping too close of an eye on what is happening on that front. However, based on the error from trying to build for Android, it would appear that Vulkan support has not yet been implemented.
