The Decanter of Endless Water, DMG pg. 161, can fire a 30 foot long, 1 foot wide stream of water that can knock a target prone. On top of that, it keeps streaming until the end of my next turn (important).
A Vampire, MM pg. 297, takes 20 acid damage if it ends its turn in running water.
If I somehow manage to get this going on a vampire that we've prevented from moving, would this count as running water?
Jeremy Crawford answered the question of what is running water. To the question:
... Is this only from natural running water, or any water that move?
Jeremy answered:
A body of running water from any source can harm a vampire.
He then clarified, when asked about water elementals:
A water elemental is a creature, not a body of running water (a stream, a river, a waterfall, or the like).
http://www.sageadvice.eu/2016/04/26/vampire-in-running-water/
So, based on that, If you can get enough water out of the decanter that it is stream or river like for your DM, then you're good. But even with the clarifications it is a DM call. I personally think 1 foot x 1 foot of moving water is enough to count as a body of running water.
I, personally, would give the vamp a dex save (DC 15??) to see if he can dodge it before it hits him.
I think you've exhausted the specific rules in your question, so you're going to need your DM to rule on this, or if you're a DM - decide!
You've demonstrated in your question that it's mechanically possible to force the vampire to end its turn within the geyser produced by the Decanter. That geyser is running water, which means that you can force the vampire to end its turn in running water and take the ensuing damage.
In the remainder of this answer, I'll explain why the geyser is running water (redundant thought that might sound):
'Running water' isn't a rules term, which means we take its meaning from common parlance. 'Running water' is a very broad term which can be reasonably used to describe many kinds of moving water, such as:
In everyday speech, the word coupling 'running water' can serve as shorthand for natural bodies of flowing water such as rivers and streams. However, the broader definition of the term is best understood by examining the effect of the adjective 'running' on the noun 'water'.
Dictionary.com lists several definitions for running, including:
The latter definition can be presumed to be derived from the former; 'running' describes the way that water flows in a stream because 'running' describes something that is 'moving or passing rapidly'. Hence, any water which is moving or passing rapidly can be considered running water.
Rules designer Jeremy Crawford does not agree with this ruling, and has said so in a tweet:
Running body of water =/= body of running water.
Jeremy's tweets are no longer official rules, so that doesn't matter. His interpretation of the rule is that 'running water' should be understood (as it often is) as shorthand for a '(presumably natural?) body of running water'. However the Vampire's stat block makes no mention of a 'body' of water, and only specifies that the water must be running. I've demonstrated above that this phrase has a much broader definition than select natural phenomena.
The '30 foot long, 1 foot wide stream of water' produced by a Decanter of Endless Water is running water. If you force a vampire to end its turn there, it takes 20 acid damage.
A DM might decide, based on their understanding of vampire-lore, their interpretation of 'running water' or their reverence for JC's design ideals, that only natural bodies of water count for the purposes of harming vampires. Rule 0 gives them license to do that: They're choosing a narrow adjudication when the rules support a broad one.
In general, damaging effects require an attack roll or saving throw. Since none are given in the item description, your DM may assign one or rule that this is not a damaging effect. Another important question is whether your character knows about running water and vampires.
