If you look at this factually.... well... I mean actually LOOK at this whole thing. Words don't do it justice.
Ringed by footprints, sitting in the moondust, lies a 2-foot wide panel studded with 100 mirrors pointing at Earth: the "lunar laser ranging retroreflector array." Apollo 11 astronauts Buzz Aldrin and Neil Armstrong put it there on July 21, 1969, about an hour before the end of their final moonwalk. Thirty-five years later, it's the only Apollo science experiment still running.
read more in a NASA blog post from 2004.
So... if we consider the fact that there currently IS a mirror rig set up on the moon for us to bounce lasers off of, this would mean that NASA scientists had planned to aim lasers at it at some point. Most likely they planned on doing it from the Earth, although who knows if they wanted to put satellites in orbit that would regularly "ping" the moon as well.
Thus, is it possible? I'd have to go with yes, since NASA has done it a few times before. In fact... here's an image of the laser at McDonald observatory in action.
Now.... I do get it that the question was about whether it is possible to do this experiment with "low powered" and inexpensive equipment. I assume that NASA has tried this with different configurations of equipment, however whether any of those configurations would be considered inexpensive and/or "low power" is up to the person reading the reports. I would imagine that it is indeed possible to recreate the experiment with equipment that would be equal to a fraction of the cost and power of the McDonald Observatory setup... as to whether anyone would consider it to still be cheap and weak is another story.
That's what I'd imagine at any rate. The question was asked before, however. And from the answer, it appears that the cost of the equipment would always bring it out of the Amateur category.