J Channel Looks Wrongly Installed And How To Fix (Water Leak)
EDIT, due to new info: Thank you very much for the further investigation! I assumed too much and should've asked if there were any other siding penetrations above.

Anyway yes, a quick and easy fix is to apply a liberal amount of almond color (whatever comes closest) caulk that's fingered or forced into any gap at just the top and sides...so any water gathered at the bottom can still drip out and breath to evaporate.

Don't fill the back of the vent with caulk. Instead use a Foam Backer Rod or even thick weather-stripping to give the caulk something to seal onto. And still, probably do 2 applications of caulk, to make sure the 3-sides are filled and adequately thick.

A proper permanent fix is at the end, but 2 materials in an optional non-removal of the vent is to either install the Drip Cap mentioned below/initially or a piece of Step-Flashing to the top of the vent, which tucks under the siding. Step-Flashing is much more customizable.

Your vent was installed with the row above as an overhang to keep water out. Unfortunately, this only works with correct J-channel picture frames (just like a window) or J-channeled penetration boxes and not with the entirely wrong vent that protrudes as much as the siding itself and has no top and sides flashing flanges.

However, you're lucky. You can just cut the siding's locking lip out of the row above the vent to get the Drip Cap or Step-Flashing flange behind the siding. You still want a tight Z-shape to your flashing, which is why I think the Step-Flashing will work out best with possibly just household scissors. You'd still caulk the sides as a wind precaution, but the top flashing would really do everything needed.

Finally, the best option is to remove the siding and old vent, which seems to be a 1-piece construction. You want a 2-piece construction that has an installation flange where the vent flaps snap onto and holds the siding to the wall...see Instructions/Assembly here, not Installation Guide here for what I'm saying https://www.homedepot.com/p/Builders-Edge-4-in-Hooded-Vent-117-Bright-White-140116774117/100673014

Further, there are also 1-piece penetration boxes that have seamless J-channel built-in (backed picture frame of J-channel to cut your own hole into for whatever accessory), but only roofing and siding suppliers have those and may not sell to an individual...definitely try online or in person though, if it's what you might want.

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I do still think you should do the below. As, water may still be running on top of the "black stuff" and will find its way in, if it hasn't started to already.

You are very correct that it's very wrong. Ideally, you should have a Drip Cap under the top J-channel. But, many installers just use the J-channel itself as the Drip Cap, since it's a little more uniform looking and of course faster and cheaper..."experts" cutting corners (literally in your case) because they know what they can get away with.

However, you can correct the inherited problem fairly easily for each window and door within the siding (I put Drip Cap everywhere possible). By, buying Drip Cap and either installing it correctly and pulling the siding panel free for a complete re-do or by cutting the Drip Cap's back off to just slip it under the J-channel.

Either way means you cut a tab into each end of the Drip Cap's bottom back corner, which is then bent down into the side J-channels to direct water harmlessly away from the top corners. Here's a great video demonstrating the proper process of J-channeling, top piece starts at about 12-min. How To Install J Channel (Vinyl Siding) https://www.youtube.com/watch?v=gmiN3BnOu3w