When water flash evaporates in space, about half the water evaporates taking almost all the heat so the other half of the water freezes to ice. The fraction of it that evaporates probably travels in all directions equally because it is a molecular level process. The velocities of those molecules will have a Boltzmann distribution, depending on the temperature of the water at each moment that the evaporation is occurring, so at first, the temperature will be warmer, but it will steadily cool as it evaporates. This will cause the velocities in the water vapor to become smaller and smaller. However, because the bolts in distribution has a very long tail, there will probably be molecules at velocities high enough to reach the moon. So some of those molecules would end up on the moon.
However, the ultraviolet light is able to break molecular bonds, and the probability of a molecule flying all the way to the lunar surface before photolysis is vanishing small. Therefore, what reaches the lunar surface will be individual atoms. These same atomic species are in the solar wind, so it will just be an in distinguishable, tiny addition to the solar wind.