Can we observe the shapes of MOs?
The very question you pose is addressed thoroughly in this open access work:
https://www.nature.com/articles/ncomms9287 . 
The short answer to your question is that the electron density can be mapped using a technique akin to diffraction as described in their work.  You mentioned a distinction between the electron density and the orbitals, and the orbital is not itself observable; but as you suggested, the electron density can be mapped.