From what I have read about the Vigilante chain, the Artificer class can replace either the Adept class or the Ranger class. So a Vigilante can be a Adept/Ranger, a Adept/Artificer or an Artificer/Ranger. But you are going to still run into same problem I mentioned earlier, finding two archetypes that will actually work well together.Vigilante chain for instance does not rely on adept or ranger class features, and in fact artificer had to be retroactively added because the Batman fantasy clearly gels with Artificer (gadgets and all), but the initial feat chain only listed two classes so they had to tack on Artificer afterwards.
The Artificer in A5e only has three archetypes (Bombardier, Machinist and Stitcher). None of which lends itself well toward a Batman-like character. So you would need to look for an O5e Artificer archetype or a 3pp A5e Artificer archetype that does. Or you would need to craft your own homebrewed Artificer for the job.
I am going to guess that Batman would have the Exalted Athlete archetype because of him training and honing his mind and body to near perfection. As for the Ranger half of this multiclass combo, I am going for the Partisan | Level Up archetype. Batman fights dirty and unconventionally against criminals in Gotham City.
Now if your vigilante makes use of a bow and a quiver filled with trick arrows (Hawkeye, Green Arrow, Artemis, etc.), then you'll be picking up the Ranger's Fantastic Fletcher | Level Up archetype. So Exalted Athlete/Fantastic Fletcher.
It's weird IMO that the A5e Artificer was retroactively applied to the Vigilante chain since none of it's official archetypes lends itself well toward that feat chain. Is there a 3pp version of the A5e Artificer that does? curious