There are a bunch which might test, or at least put some constraints on, LQG (Loop Quantum Gravity).

In a nutshell, very distant point sources will be 'smeared out' by space (which is a 'spinfoam' in LQG), so determining the angular size (as we detect it) of very distant point sources should also test (or set limits on) some properties of space.

What might these very distant point sources be? Supernovae and GRB afterglows are the two which spring to mind, but there might be others. How to determine angular size? With the ACS on the Hubble, for one; maybe VLBI observations as well, and perhaps one or more of ground-based optical interferometers. With the right amount of good luck, a night-side lunar occultation might also be possible.

The role of amateurs? Little chance for using their own telescopes I suspect, but all the Hubble raw and processed images are public domain, so maybe a re-examination of archival supernovae images, with the right kind of image analysis software?