Yes, you can use a signal mirror by moonlight!
My teenage son and I just got back in the house, it is -20*c out with about 40cm of snow on the ground (clear and cold with lots of ground reflection). The moon is well up in the sky in the southeast and is almost full (greater than 95%), it is casting heavy shadows.
We used a 3"x5" glass 2 layer sighting mirror (labeled - Mirror, Emergency, Signaling, Mark 3, Type 2) that I have had for at least 20 years.
Trying to use the 2-hand technique was difficult as the mirror did not return a lot of light, but using the sighting hole/fireball method it was very effective, maybe even easier than in the daytime. The bright spot was easy to see in the centre of the mirror and simple to aim. We only had about a 30m cleared area to shoot the light signal but the flash was very bright at night and I am sure it would get someones attention at quite a distance.
It was very difficult to use the 2 hand technique because I could not track the location of the reflected beam, even when I knew I was bouncing it off a wall 30 metres away. I think this just has to do with the limited amount of light you are reflecting. A signal mirror that you can aim is much more functional at night.
Well you learn something everyday, I think my neighbours learned (again) that we do strange things at our house?
Mike