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Can they take advantage of their rule to shoot characters or other rules of theirs such as death is not enough, harvester of torment, etc.?

No? The rules don't say they work when they are embarked, so they cannot. Rules need to explicitly state they affect embarked units for them to affect embarked units. That is exactly what "Unless specifically stated" means.

I agree with your RAW interpretation. The reason I posted this question is that - I admit I could be mistaken - it appears to me that the rule under consideration intends to prevent embarked units from benefiting / being affected by auras (or granting them to nearby units) as well as using stratagems on them, etc.

Not knowing the game very well... does the possibility of embarking a vindicare assassin on an open topped imperium vehicle to snipe characters exists (assuming he could target characters while embarked)?

Additionally, death is not enough, humbling cruelty, harvester of torment, etc. are abilities that the DJ alone takes advantage of, he doesn't grant them to nearby units.

I am all but certain there is no open topped vehicle in the game a Vindicare assassin can embark in.

There's been some debate about exactly what "normally" means, in the context of this.

Extremists posit that literally only the lines of text that explicitly say they apply can be used. But most reasonable people read abilities that come into affect when performing an explicitly allowed action as implicitly applying.

An example is plasma weapons, with the rule that the firing model is slain when overcharging and rolling a 1. It is implicit that this rule applies when embarked and shooting the weapon, provided that explicit permission has been given to shoot in the first place.