The rules, in this case are pretty clear. The spell never mentions mirrors or reflections, it “summons illusory duplicates." Spell names aren’t indicative of their mechanical effect. See Chill Touch.
There’s also vampire wizard statblock that has Mirror Image on its spell list.
It would be funny if the spell just failed though.
Now I want to know wether the rules allow you to stand perfectly inside your mirror image (don't have a 5th ed./any other D&D ed. rulebook to check myself, sorry)...
Because if it does (without going all wavy or otherwise distorting) and it casts a reflection this could be a fun "find the hidden vampire in this banquet hall with mirrors" scenario!
I think the illusory duplicates made by the magic are designed to "move around in your space" to make it hard to tell which one is actually you - the purpose of the spell is to make you harder to hit, so it's behaviour probably works to let that happen. It'd also be very difficult to get away with it in a social setting because you have to cast the spell first, it makes four duplicates, and they don't last that long...
Now on the other hand... Trickery Cleric "invoke duplicity" allows you to control the movements of the duplicate (which is incorporeal), so it might be possible to hide it inside you if you were really practiced. Konsi sometimes summons it in her exact position while standing still, then uses her second channel divinity slot to turn invisible and run away.
It'll only give you cover for a moment, but if they're doing some sort of "check every visitor for vampirism" check on the door, it might get you past.
My ruling would be that the vampire does not appear in the mirror but the illusions do. If someone got creative I would also rule that if you aim your attacks based on looking at a reflection you would be attacking a single target with full concealment (normal attack roll with disadvantage) and not rolling to see if you hit an illusion because you're aiming specifically at where they aren't.
That's based on the idea that vampires are weak to silver, right? In D&D, they don't seem to have that weakness. It should be werewolves, devils, wights, jackalweres, wraiths and night hags that don't have reflections in silver mirrors. Also, the mirror in equipment is steel. The only way to get a silver mirror is to learn Sanctuary and have a component pouch.
I've read something along the lines of it being because silver is a "noble metal" (or something like that) and vampires are unholy/foul creatures, somehow making them not be reflected by silver mirrors
It's a common trope in media and fantasy lore, Quite a few tropes of vampires are listed in the flavour text, rather than the statblock, and this is one of them.
I suppose it depends on whether applying the mirror to a vampire with mirror image counts as an "attack." It is an action that functions similarly to a divination spell. If the Vampire casts Mirror Image, and then you try to "target" the vampire with the mirror, there's a chance you target one of the mirror images, and it would be visible in the mirror.
Vampires also don't cast a shadow, and Mirror Images are identical images, so none of them would cast a shadow. So while the mirror trick might not work against a vampire who has cast Mirror Image, you would be able to use any light source, including ambient light, to check if any of the images are casting a shadow.
I don't I can make a good case for "looking in a mirror" to use the "attack" rules. If it's an attack you'd have to make an attack roll (if you're not making an attack roll, it's not an attack.)
Likely, if there was a d20 roll involved, it'd be a perception check.
You're not wrong about the RAW, but it's a point of contention for the RAI on how Mirror Image affects non-attacks that target the caster.
My feeling is that using a mirror on an enemy is like an attack for the purposes of interpreting the RAI on Mirror Image. Mirror Image only specifies what happens if the caster is attacked (the false images have a chance of being targeted instead) but there are Divination spells and some offensive spells like Magic Missile that require no attack roll. Some people interpret that to mean that those spells ignore Mirror Image, but that argument doesn't track for me.
Anything that requires targeting an individual by sight should be affected, even if it isn't technically an "attack" per the rules. It should be up to the DM to decide. The description of Mirror Image makes it clear that it confuses anyone who can see it, making it difficult to determine which image is the real one. It describes specifically how to calculate attacks, but it also doesn't specifically say that non-attacks are unaffected.
If you use that interpretation, it would suggest that pointing a mirror is like targeting the character who has cast Mirror Image. You would have to roll a d20 and compare that to 6, 8, or 10 depending on the number of mirror images. Perception can't see through Mirror Image.
You might see no reflection (which would confirm they don't cast a reflection), or you might see the false image's reflection (which would be ambiguous).
This all assumes you're using a steel mirror or similar item to see the reflection. If there is a mirror in the room, you would be able to see more of the room, and presumably you'd be able to tell if the caster had a reflection by comparing the number of reflections you see to the number of duplicates in the room with you.