People constantly ask "explain my magic via science!" questions on this site. The only problem with the idea is that the entire purpose of science is to dismiss magic and disillusion humanity.
It's impossible to explain magic via any means other than magic, but some people have to learn the hard way, so here goes.
Your hapless vampires must possess two qualities in order to not cause a reflection in a mirror.
The light reflected off their bodies must, for one reason or another, not be reflected back (at least in a way that we would expect... hold that thought).
Their bodies must also be capable of passing photons from behind them so that the area behind them is reflected, rather than themselves. In other words, for one reason or another your vampires are transparent (at least in a way that we wouldn't expect... hold that thought).
Answer #1
@Trioxidane is, of course, completely correct. What you're asking for (a scientific basis for one aspect of the vampire mythos) is impossible. For the very same reason that vampires, themselves, are impossible. Science doesn't allow it. But this is boring, so let's ignore it.
Slightly greasy solar atoms
There's a reason experienced and successful authors don't explain the gory details. They're irrelevant. This aspect of storytelling is known as Checkov's gun. Simply, don't explain anything that isn't relevant to your plot because if you do, you're distracting your reader from the story you're trying to tell. From Checkov's perspective, don't tell the reader there's a gun hanging over the fireplace unless that gun has a role to play later in the story.
And that's where slightly greasy solar atoms come into play. I'm a huge fan of Luc Besson's The Fifth Element During that movie it's necessary to reconstruct Milla Jovovich's character from a single cell. Do they explain it? Heck no! Most of the process is relatable to the audience as a far-future form of 3D printing, but skin isn't just flesh... it's an organ. How does Luc explain flesh growing over the musculature of Milla? "Slightly greasy solar atoms" are introduced to "force the body to protect itself" by growing skin.
Insofar as science is concerned, what Luc came up with has nothing at all to do with science. It's just some really cool technobabble that justified "what needs to happen next." So, rather than scientifically explaining a complex piece of machinery that had no purpose anywhere in the story other than in that one scene, Luc opted to simply hand wave it. Good for Luc!
And he's not alone. When J. Mills Goodloe and Salvador Paskowitz wrote the script for The Age of Adaline they explained Blake Lively's character's inability to age this way:
At 8:55 a bolt of lightning struck the vehicle discharging half a billion volts of electricity and producing 60,000 amperes of current. Its effect was threefold. First, the charge defibrillated Adaline Bowman's heart. Second, she was jolted out of her anoxic state causing her to draw her first breath in 2 minutes. Third, based on Von Lehman's principle of electron compression in deoxyribonucleic acid - which will be discovered in the year two thousand thirty-five - Adaline Bowman will henceforth be immune to the ravages of time. She will never age another day.
It's utter nonsense! But it sounds sciency, right?
To close or not to close the question
And that brings us to the problem with asking for a scientific explanation of magic. Do we close the question or not since the only response must be technobabble that has as much relevance to science as the lyrics of "Macarena" do to pancakes. So... at the risk of supporting a question that probably should have been closed as opinion-based or too story-based and at the risk of having this answer voted deleted as a lengthy comment, let's throw a nearly meaningless explanation (from the perspective of science) at you and see if it sticks. And if I'm REALLY lucky, you'll learn something about why it's insanely important how and why you ask questions here (because we're not here to help you write a story, we're here to help you build a world, thus "scientific sounding terms" isn't something we do...).
Answer #2
If you're fortunate enough to live long enough to look at a vampire... to really look at a vampire... you'll see that they're slightly blurry. That's because they're vibrating. It's why a stake anywhere but the heart doesn't kill them. They have more in common with a cloud than they do humanity. What does this vibration buy us? Some light passes through them. Just enough to make the reflection in the mirror look convincing... unless you really look at the reflection and realize that the vampire-shaped area isn't quite as convincing as everywhere else.
The light that bounces off the vampire, which would normally be reflected in the mirror, becomes, shall we say... wiggly. It's the vibration, you see, which causes the photons to go haranguing off into space in all kinds of ways and with all kinds of twists and spins. This isn't much of a problem for the human eye (unless you really look at the vampire) because the human brain is in the business of processing images into a comprehensible format. It's why you can see the shadow of your hand in a darkened cave. You're brain is perfectly capable of showing you the vampire! It's basically using the same capabilities that cause hallucinations — just in a useful way. But! When the light winging all over the place hits the mirror, combined with the natural imperfections of mirrors, the light is spread all over creation. So little comprehensible light can reach the human eye in such a case that there's nothing to use to recreate the vampire.
Your real problem isn't the vampire... it's the vampire's clothes! Those would be seen in the mirror just fine no matter what the theoretically science-based vampire can do. So, in your world vampires have neither scruples nor modesty. They wander around buck nekid so you can't see them in a mirror. (And most people think the vampire's ability to hypnotize people has something to do with hypnosis! Hah! It's really simple shock. Or awe. Kinda depends on which side of the fence you and the vampire are on.)
You know, you could increase the mass of the vampires to the point that the light behind them warps around them so nobody can really see them... but that would require the Earth to implode and time to stop, so we'll pass on that solution.