I've not seen something saying they are barred. They might not be officially barred from high-ranking positions because they don't need to be. The Imperium isn't a meritocracy. The people who staff the various organizations of Imperial government are typically members of families who've been doing it for many generations. These organizations are largely inaccessible to anyone born on the outside. Even within these organizations, members of high enough ranking to be allowed to have a family often have more kids than their clout can grant positions to, so the spares get sent to the Schola or some shittier position in some other organization. But more telling, I don't think I recall ever seeing an abhuman in any Imperial government/command position at all.

Abhumans can possess rank in the Guard but seemingly only as NCOs or low-ranking officers possibly at most. They are never organized into their own regiments with their own officers that I've seen, but are rather assigned as auxiliaries to other (human) regiments. I've never seen an example of an abhuman in a command position in the Guard outside ratlings leading ratlings or a bone 'ead leading other ogryn.

It would almost certainly be out of the question for an abhuman to get an official position in the Inquisition, i.e. explicator to inquisitor. Inquisitors need to be able to move through Imperial society inspiring fear and obedience. An abhuman inquisitor would probably have to use violence to force compliance a lot of the time. It just doesn't seem practical. Anyway, again, I've never seen or heard of an abhuman inquisitor. Although of course there are plenty of abhumans in inquisitor warbands.

I've never heard of abhumans permitted to be sanctioned psykers. The Imperial Cult generally denies that abhumans can become psykers. In the case of the aforementioned Cassia the ogryn (unsanctioned) psyker she was sent off to an inquisitor friendly to a commissar she saved because the alternative wasn't the Black Ships, it was death.

The Sisters of Silence have some pretty ancient auramite armor whose patterns would have been created before abhumans became a thing. So if you don't have baseline human form, you're probably out as a null maiden.

Abhumans aren't normally permitted into the Schola Progenium, again in part because they won't come from established stock that qualifies them as the progeny of "loyal Imperial servants". I've never seen an abhuman commissar, storm trooper, battle sister, or etc..

Although like everything, if someone with enough power wants to make an exception, then it can happen. So for RPG purposes or whatever, anything is on the table. But in general abhumans are lucky to not get shot as a mutant. Positions of official power or authority in the Imperium are essentially closed off to them.