In any case, I think you have the right idea. A liquid #"boils"# when its vapour pressure is equal to the ambient pressure, and bubbles of vapour form directly in the liquid. The #"normal boiling point"# is specified when the ambient pressure (and thus also the vapour pressure of the liquid) is #"1 atmosphere"# (of course now we're down here on Earth).
At #100# #""^@C#, water has a vapour pressure of #1*atm#. The temperature of the moon varies between #-150# #""^@C#, and over #100# #""^@C#. However, the ambient pressure is much, much lower than on Earth. Even on the dark side of the moon, water is likely to exist as vapour (even ice has a vapour pressure). This site estimates that the pressure on the moon is #3xx10^-15# #atm#. Lunar water is thus likely to exist in the gaseous phase.
