Late to the party but I just watched a video by Scott Manley from August 2020 discussing the exact question, in the context of the TV Series "For All Mankind" which features Shuttles outside LEO.
Manley's tenor is that it's not principally impossible to reach the moon in a Shuttle, just impractical; the shuttle simply is not the right vehicle for the job, with wings and wheels and stuff.
In order to make that possible one would want to use the main engines, which should in principle be able to restart in zero g, after an ullage burn (thanks to OrganicMarble for explaining a video to me that he didn't even watch!) by the orbital maneuvering system (OMS) if the tanks are pressurized and electricity is available.
As others have pointed out, there is a lot of delta-v involved. For that, you'd need more hydrogen and oxygen than fits the cargo bay. Manley suggests to bring the external tank to LEO instead of ditching it at 60 km and refuel it there with hundreds of tons of fuel from some handwave source, and then fly it to the moon (into an orbit, that is). The nice thing is that you have enough spare room to bring reserves for evaporation etc.
The largest obstacle also in Manley's opinion is the high energy at re-entry into Earth's atmosphere. Re-entry would have to be done without the tank, so the main engines cannot be used for an entry burn. Manley does not discuss using the OMS thrusters for that; maybe they don't provide enough thrust to sufficiently decelerate the vehicle.