Also, technically the Vampire doesn't have to die if forced to resume his normal form while a 0 HP. He's unconscious but the DM can allow him to make Death Saving Throws. If the party doesn't finish him off (just like when he's in the coffin), he can turn back into mist once the moonbeam is gone.
The idea is to hit it with
moonbeam and then bring it to 0 hit points (either with the
moonbeam's own damage, or some other way). When the vampire hits 0 hp, it would normally trigger the Misty Escape ability: When it drops to 0 hit points outside its resting place, the vampire transforms into a cloud of mist (as in the Shapechanger trait) instead of falling unconscious, provided that it isn't in sunlight or running water. If it can't transform, it is destroyed."
Moonbeam, assuming the vampire failed its save (and by the time it's brought to 0 hp, it is usually out of Legendary Resistances, and it has disadvantage on the save), prevents the vampire from transforming. Thus, it is destroyed. This seems fairly clear by the book.
Now, you could also use
moonbeam on a vampire that had already fallen to 0 hit points and gone misty. This is a bit more iffy. For one, the Misty Escape trait says the vampire
can't revert to humanoid form, and the
moonbeam spell says it has to. Also, it doesn't
drop to 0 hp by being hit with the beam, so the first paragraph doesn't trigger. I think I would judge in favor of the
moonbeam here - it seems weird that it would kill a vampire if it hits it before it goes mist, but not afterward. On the plus side, it wouldn't have disadvantage on the save if hit in mist form, because the advantage it gets on Constitution saves cancels that out.