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.
But do their mirror images show in a mirror, as the mirror image itself isn't a vampire?
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.
...All of this is pretty sketchy though, it's definitely pushing the limits of what these spells or abilities are intended for, and "having your image move around inside you perfectly so nobody notices" would require an extreme level of control that I imagine most, if not all casters wouldn't have.
If we're trying to design an NPC vampire to foil the mirror check, it's probably better to have them use a custom spell or magic item that's explicitly for this purpose than to try and repurpose an existing spell to do it. "This amulet creates an illusory duplicate of you that fills the same space as your body. When you turn invisible, the illusion remains visible and moves in a random direction away from you" for example.
This approach has a lot of advantages I think.
There is a spell for these types of shenanigans and it's called Mislead
It's a classic. I wish the movement didn't require your full action though - the spell is level 5.