Could this be done technologically, assuming the mirrors are completely normal ones and there are no psychic powers in this universe? Maybe, with one of these options:
First, take a look at this article a hypothetical near-future invisibility suit. The suit would be covered with tiny omnidirectional light sensors and light emitters, all in contact with a computer. A diagram of such a sensor/emitter pair from the article:
To achieve true invisibility, optical camouflage must capture the background from all angles and display it from all perspectives simultaneously. This requires a minimum of six stereoscopic camera pairs, allowing the computer to model the surroundings and synthesize the scene from every point of view. To display this imagery, the fabric is covered with hyperpixels, each consisting of a 180 x 180 LED array behind a hemispherical lens.
When a certain wavelength and intensity of light strikes a sensor in a particular direction, the computer can calculate the trajectory that light ray would take along a straight line if the person were totally transparent, and where that line would intersect the opposite side of the body. Then, it can instruct the emitter at the intersection point on the opposite side of the body to emit a light ray in the same direction with the same wavelength and intensity, which we might call a "masking ray". If this is done at every point along the body for light rays coming in every direction (even if it's not perfectly fine-grained so the emitted light is a bit pixellated), you can have a form of practical invisibility.
I think it should be possible to modify this so that if you look at the person in the suit directly they are visible, but they will appear invisible in mirrors. My idea here is just that the image-processing software interpreting the data from the sensors can distinguish human eyes, and once the suit has identified all the human eyes in its surroundings, it can selectively instruct all emitters not to send masking rays in the direction of any human eyes. You may also want to have the software be able to distinguish camera lenses if you want the vampire to be visible on camera, and also be able to distinguish the eyes of various kinds of non-human animals along with humans (although you might not want to do this if you want these technological vampires to creep out animals because they can hear and smell them but not see them).
The only problem with this is that if the software sees someone's eyes in a reflective surface like a mirror, and interprets that as another set of eyes, then if it fails to send the masking rays in the direction of the reflected eyes, the person will be able to see them in a mirror. But with sufficiently sensitive sensors and smart software it may be possible to differentiate reflections of eyes from regular eyes which lie directly in the line of sight of the suit, since any real-world reflective surface is going to have some combination of specular and diffuse reflection (also, the software could be continually trying to build a model of the environment based on sensory information, so it could notice when the image on a surface seems to be a duplicate of some mapped region of the environment in the opposite direction).
Also, note there would be one side effect of this technology which isn't usually part of modern vampire lore (though it was apparently a feature in some older vampire myths): besides having no visible reflection, the vampire will also cast no shadows. I suppose if you want to avoid this, if the software can distinguish eyes in mirrors and other reflective surfaces from regular eyes directly in the line of sight, it could be designed to only send the masking signal in the direction of reflections of eyes, and then it would have a normal shadow. But this would have the disadvantage that if the system failed to notice a pair of reflected eyes because the reflection was far away, distorted, and more diffuse than specular (say, a faint reflection on a dull metallic teapot) then a person near the reflected surface could notice the vampire, whereas sending the masking signal everywhere except non-reflected eyes would avoid this error.
This option has the drawback that it requires the vampires to not only have this sort of detector/emitter system lining every surface of their body, but also to have it lining the surface of whatever clothing they wear. But if you imagine the vampiric "infection" was created by some civilization with advanced technology in the past (perhaps native to Earth of that time, or perhaps involving aliens or time travelers), it could involve a swarm of nanobots that envelop each new vampire, and it could be smart enough to form an envelope that extends to clothing as well as the vampire's body, with the nanobots forming tiny light sensors and emitters on the surface of both. Such nanobots could perform other plot-relevant functions like repairing damage to the body (aside from stakes to the heart and beheadings), preventing decay and keeping neurons and muscles functional even if the cells are not really alive, distinguishing sunlight from other forms of light and causing an explosive reaction when sunlight is detected, etc.
If you let the tiny robot swarm get very advanced, perhaps involving femtotechnology rather than nanotechnology or perhaps some particles unknown to current physics, there could also be an alternate solution that doesn't involve selectively detecting nearby eyeballs. Imagine the tiny machines absorb all incoming light, but the machines are designed to mimic the optical properties of the bit of the vampire's body or clothing directly underneath them, sending out light-like signals that match whatever light the bit of body/clothing underneath would emit under the same lighting conditions. However, instead of the emitted signals actually being light, imagine the signal is itself a subatomic-scale machine, or perhaps an exotic particle unknown to current physics, sent out on the same path the light ray would have taken (if you want to bring exotic particles in, I suppose you could ditch the idea of tiny machines and imagine the vampire's body is surrounded with some kind of field which is also unknown to current physics, which has the property that any photon exiting the boundary of the field is transformed into an exotic particle with the same momentum and energy).
Being smaller than an atom, this machine or exotic particle (which I will abbreviate M/EP to avoid having to keep saying 'machine or exotic particle') is designed to mimic the behavior of light in the sense of passing through transparent materials like water or glass, and even having its path refracted by them. If the M/EP comes to a bit of matter that would absorb or scatter photon, like an electron in a photosensitive molecule in the human retina, then we can imagine the M/EP imparts the same amount of energy and momentum to the bit of matter as that bit of matter would have gotten from the photon (if it's an exotic particle we can just imagine the particle itself has the same energy and momentum as the corresponding photon, if it's a tiny machine we can imagine the machine emits a photon of that energy and momentum when it detects a bit of matter that would naturally absorb or scatter a photon). And if the molecules in your retina react the same way, you will be able to see the vampire if you look at it straight-on.
But assume the M/EP behaves differently than a photon would when it encounters to some bit of matter that would reflect a photon--at this point the M/EP will just get absorbed or otherwise destroyed by that bit of matter, rather than travel on a reflected path (if it's a tiny machine it could just break up at that point). This will ensure that no M/EP can take a path that goes from the vampire to a reflective surface and then to a human retina (or camera lens or animal retina). Finally, if the M/EP encounters a bit of matter that has some probability $P_1$ of reflecting a photon, like a piece of glass which reflects some incoming light but lets the rest pass through, then the M/EP particle should have the same probability $P_1$ of being destroyed/absorbed, and otherwise will interact with this bit of matter the same way a photon would. This ensures that even though a person's eyeball is partially reflective, the photosensitive molecules in the retina will be imparted with the same energy/momentum by the M/EPs as they would get from photons if the vampire wasn't surrounded by this advanced technology and was just emitting light normally. But even if you were standing right next to a vampire and looking closely into the eyes of another person standing across from both of you, with this technology you wouldn't see the vampire reflected in the person's eyes (this wouldn't work with the more realistic invisibility suit technology I discussed in option 1). Also, with this method the vampire will cast a shadow normally, and the shadow will even extend over reflective surfaces like water even though the reflection itself will be absent.
Of course, this second option is well in the realm of technology indistinguishable from magic, so if you're going to pick a solution of this kind, it might be simpler just to assume that our apparent physical world is a Matrix-style hyper-detailed simulation, and whatever beings are running the simulation have just programmed in special alterations to the laws of optics for light rays emitted or reflected by vampires (an option already mentioned in user100487's answer).