I don't think you can link shared objects into a static binary because you'd have to patch all instances where the code reads the PLT/GOT, but this can be arbitrarily mangled by the optimizer, and turn them back into relocations for the linker to then resolve them.

You can change the rpath though, which is sort of like an LD_LIBRARY_PATH baked into the object, which makes it relatively easy to bundle everything but libc with your binary.

edit: Mild correction, there is this: https://sourceforge.net/projects/statifier/ But the way this works is that it has the dynamic linker load everything (without ASLR / in a compact layout, presumably) and then dumps an image of the process. Everything else is just increasingly fancy ways of copying shared objects around and making ld.so prefer the bundled libraries.