The Wall of Water spell doesn't specify whether the water is moving, only that it is difficult terrain. It is 1' thick and thus may or may not qualify as a sufficient quantity of water to cause the vampire to be 'in running water' if he is restrained while in it.
So the question is, would holding a vampire in the area of a Wall of Water damage the vampire?
As you point out, the spell itself doesn't specify whether the water is moving. If it was moving, it would likely be specifically called out in the spell text.
For example, Tsunami, a similar water creation spell, states
A creature caught in the wall can move by swimming. Because of the force of the wave, though, the creature must make a successful Strength (Athletics) check against your spell save DC in order to move at all. If it fails the check, it can’t move.
It also explicitly pushes creatures around. Neither of these abilities are enumerated in the text for Wall of Water.
As a side note, you could potentially get the water moving by casting Shape Water on the resulting wall of water. It might be a bit controversial, but I'd rule that it would work...
The way i see it, it's a solid wall of water, there would be no reason for it to move, think of it like this, if i were to need to create a wall, would it need to move to hold something back? due to the nature of the water wall, the default would be unmoving, and since it doesn't specify moving, it is likely unmoving.
