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.
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.