I'm simply curious of the Adeptus Mechanicus' view on mutants and if they're allowed in their ranks.

Short awnser - No.

Long awnser - Extreamly unlikely. Mutants tend to simply get put down at birth or hide in exile, if they're born anywhere remotely you'd call "Imperail Civil". That being said, it dose depend on the mutation (A tech preist with minor mutations might cunningly use his access to bionics to hide the fact he has mutations). If it was caused Later down the line, after he became a tech preist, then joining the ranks of a Rogue Trader's fleet sounds like a very logical coruse of action. Rogue traders tend to have a less "burn the mutant" approch, espcailly if the mutation would be advantagous.

Ship needs more than that one Tech-priest. But yes, it should be easier to hide and dodge gene-scans either in a sleepy province or where people are concerned with other things entirely.

If you already were a mutant, I think they might be iffy, but they'd look you over. Depending on what the mutation is, it could easily be excised, and replaced with cybernetics; something the AdMech rather intend to do to the "good" parts of you, anyway. On the other hand, the AdMech work around numerous sources of possible mutation, so they can't really be too surprised when it pops up. Look at the people who live on Mars; they are all irradiated, burned out monsters, wearing masks just to breathe, putting even Kriegers to shame, because they live on a Forge World, and then an especially irradiated one, even by those standards, and the baggy, red robes can help hide things, too.

If you already are a Tech-Priest, you might think that all the tech will make you resistant to this problem, and, to a point, you would probably be correct, but one must remember that, as much as many of them look all-mechanical, for much of their careers, most will still be decently organic; they don't get a new upgrade every week, and even they seem to as much make an achievement/reward system out of it, instead of just building robots, and putting brains in them, or making you mostly metal by week 2, to please the Omnissiah, when they do have the resources for that. Again, with the leaky reactors, nuclear fallout, and warp energies many will be exposed to, any of that organic flesh might sprout another arm, or something.

I don't think they'll be as bad as others, if for nothing else than because they think they can do something about it; lop off the offender, and replace it with cold, eternal metal. If you aren't a terrible mutant, they can probably "fix you", if you will, and their attire can conceal many abnormalities. You just had better not revel in you mutation, or draw too much attention by it, that it makes the organization look bad. As disposable as the average person is, in 40k, the type capable of being a Tech-Priest, maybe even someday a Magos, is much more rare, and thus, much more worth it to look past a few things; they were going to change you, drastically, anyway.

Explorator fleets might well tolerate it if the individual showed talent - once a priest mutation is unlikely to be a huge issue unless you get a load or really terrible ones.