Actually, Servitors are all lobotomized which would prevent them from having the mental capacity and force of will to use psychic powers at all. If you could have psyker-Servitors, I can assure you that the Imperium would have no other kind of psyker.
Also, the only profession allowed to psykers in the Imperium is full-time Psykana trained psyker. Anyone found at a reasonable age by the Black Ships gets carted off to Terra to either be fed to the Astronomican or maybe, if they're really good and really lucky, getting trained. If they find someone who's been hiding their powers? Dead. A latent psyker demonstrates gaining the ability without the benefit of an extremely powerful patron (say, and Inquisitor) to get them sent to train and be sanctioned? Also dead. Since you literally can't join the Mechanicus without being checked up on by the Black Ships, you can't have a Mechanicus Psyker. You could, perhaps, have a Psyker who was a Mechanicus Acolyte for a while under very specific circumstances, but never someone who is both at the same time, and a high-ranking Adept would pretty much certainly be killed for tainting the perfection of the Machine-god's followers.
Edit: Space Marines tend to view spontaneous development of psychic powers as evidence of Warp corruption and put down the unfortunate, but there's a lot more room for deviation and variance here. Grey Knight Techmarines proves that it's not entirely outside of the realm of possibility for other Chapters to have psychic Techmarines as well, but expect it to be vanishingly rare at best, because both Librarians and Techmarines require years to train and a great deal of special attention.