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In my last session as a player, me and the rest of the party found a bunch of vampire spawns, and at the start of the battle, the group's druid cast Tidal Wave.

Me and another friend at the table, we both experienced DMs, remembered "hey, that's running water!" and got excited with it. The DM at the table agreed with our claim, and the water conjured by the druid spell was able to trigger the vampire weakness to running water, as the spell is unclear of when the water vanishes (in terms of turns, rounds, etc).

The space where we fought the vampire was reasonably small and "sealed", so there wasn't that much space for the water to escape after the spell was cast. The location where the fight happened was (Curse of Strahd spoiler ahead):

the room full of vampires at the Coffin Maker Shop in Vallaki.

Can the Tidal Wave spell trigger a vampire weakness to running water? If not, why?

I think most people would probably agree that Tidal Wave counts as running water, and there's no definition of running water in D&D 5e to prove them wrong. More importantly, your DM agreed, and its their opinion that matters. So Tidal Wave does count as running water. Unfortunately, for the most part it doesn't actually matter.

Tidal Wave has a duration of instantaneous, and ends with this sentence:

The water then spreads out across the ground in all directions, extinguishing unprotected flames in its area and within 30 feet of it, and then it vanishes.

Importantly, the "and then it vanishes" was not present in the original Elemental Evil Player's Companion, which might be why you weren't sure how long the water lasts.

So the vampire will only be in running water at the moment the Tidal Wave spell happens. Therefore, there's no way for Tidal Wave to trigger this:

The vampire takes 20 acid damage if it ends its turn in running water.

Similarly, there's no way for Tidal Wave to trigger this:

The vampire regains 20 hit points at the start of its turn if it has at least 1 hit point and isn't in sunlight or running water.

However, there is some good news:

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.

So if you reduce a vampire to 0 using the Tidal Wave spell, it will probably be destroyed instead of escaping to its coffin. Obviously, this will require some pretty serious metagaming (or just getting really lucky).

Personally, I would side with the Rule of Cool and allow Tidal Wave to trigger the weakness. However...

By RAW, the combo is impossible.

The tidal wave spell description says:

The water then spreads out across the ground [...] and then it vanishes.

Regardless of real world concepts like velocity or drainage, the wave appears, crashes, spreads, and magically vanishes, all in an instant.

The vampire's "Harmed by Running Water" weakness says:

The vampire takes 20 acid damage when it ends its turn in running water.

There is no water remaining at the end of their turn. Thus, they can't be damaged by tidal wave in this way.