What do vampires have to do with the phase of the moon? I've seen a few animes that linked a vampire's powers to the phase of the moon. Where does this idea come from?

Are there actual Japanese myths about vampires that this stems from? Or something odd that got mixed in when the Western myth was translated into Japanese? Something else?
Full moons are traditionally 'spooky' and are often associated with mystical creatures from the 'darkness'.

Midnight during a full moon is called the Witching hour and is when these many of these creatures gain their full strength.

An obvious example of this is werewolves who transform under the full moon, but witches and other beings are affected also. A Halloween night (or Samhain as it was called in Ireland, where Halloween originates) with a full moon is when creatures abilities are at their most potent. 

It is thought that this association was caused by the reactions that normal animals have with changes in the moon. (Birds get restless, dogs bark a lot, etc - much like how animals can predict earthquakes)

Bringing this back to vampires in particular, they also are affected by the phases of the moon. A full moon (and being well fed) will mean that their strength is at their fullest, whilst a moonless night could influence their desire for nourishment through blood. 

You should also remember that in many tales of vampires, they cannot go out in the sun. The association could stem from simplifying the moon to be the 'opposite' of the sun.

TLDR: The moon is often used as a general source of power for many mystical creatures. But vampires are also known (outside anime) to be associated with the moon.