In Quantum mechanics, position is an operator while time is just a parameter. In Quantum Field Theory, time and position are both parameters.

 

For a successful Quantum Gravity theory, shouldn't space and time be operators? Have there been any notable attempts at a quantum theory with space and time as operators instead of parameters?

LQG is the best bet right now.

 

String Theories are basically screwed as they agree completely with SR, and are background independent anyways.

 

It'd be real nice to have a fresh new idea pop up sometime though.

 

Start thinkin'!

That is a problem. In any canonical approach one needs to cut space-time into space and time. This is the origin of the problem. Time usually plays a distinguished role. Even in quantum mechanics, time is not a hermitian operator.

 

That is why path-integrals are the best way to proceed with a relativistic quantum field theory.

 

Something related you might be interested in is multisymplectic geometry. It is the geometric setting for covariant Hamiltonian field theory. The idea being to formulate a classical theory with does not use the space-time cut. However, as far as I know due to the issues with Poisson brackets no canonical or deformation quantisation procedure exists.