Could plants do photosynthesis at moonlight?
Of course they can and do, except in total darkness (spectroscopically, only bands in the far red and in the blue spectra matter - blanking these affects 'total darkness').

In photosynhesis a photon is adsorbed by Photosystem II to break down water into oxygen and protons in solution. Another photon must be adsorbed by Photosystem ! to power the enzymatic machinery that make NADPH and ATP that power the Calvin cycle. 

We know the sun powers photosynthesis effectively. The sun produces somewhat less than 100,000 lux (lumen per square meter). Moonlight is one one millionth of this or about 0.1 lux. One lux is something like 10^15 photons per second (per square meter); so moonlight provides something approaching 10^14 photons per second to drive photosynthesis. The spectrum of moonlight is not markedly different from that of sunlight. So, moonlight provides an ample number of photons per second per square meter to power photosynthesis.

The trouble, though, is that the rate of photosynthesis is low compared to the rate of metabolism in the rest of the plant. So, in effect, the plant gives off carbon dioxide by night and oxygen by day even though both gasses are being emitted all the time. 

EDIT (per request) some references:


Photosynthesis
Respiration
Moonlight lux versus sunlight
Photons per second in a lumen