LibGDX on Linux depends upon the LWJGL library for setting up rendering support. So, what you want to find out is if/how LWJGL can run without a GPU. Both LibGDX and LWJGL expose fairly low-level OpenGL commands, so I think what you need is the Mesa library (it exposes OpenGL support, but does everything in software).
From this question:
Does LWJGL (bare) use software or hardware rendering? it looks like you can ask LWJGL to not require hardware support. But, I think you still need some software-support enabled (and I believe that would be Mesa).
If you get this stack working, I think the next step is to have your "game" render to a FrameBuffer and then use Libgdx to render the FrameBuffer to file inside libgdx or OpenGL. (There may be ways to get lower layers of the stack to render to a file, too.)