The in-game shipping of activists or immigrants to different locations to coincide with plans, spreading of propaganda, that sort of thing. It is a concept a bit similar to the various humanitarian corps that have existed throughout the years, but instead of humanitarian it is attempting to be a political machine.
This made me think of the Sisters of EVE. You did state that you wanted something other than humanitarian though.
Sisters of EVE is interesting in that they are widely accepted by all factions. Even the Blood Raiders have an interest in them, referring to them as 'little hunters' and following them around. They also have their own ship types, cool looking cloaky probey ships.
'The in-game shipping of activists or immigrants' makes me think of the Amarrian underground railroad organization for liberating slaves. I think it's called 'Ladder of light' or something? The name is mentioned as part of an Amarr three part mission.
Slavery is a pretty central issue in EVE. The Amarrians defend it, the Minmatar despise it, the Gallente oppose it, and the Caldari think it's obsolete.
As far as roleplay goes, you could buy freed slaves and roleplay taking them to a retraining facility to become skilled free labor in Minmatar, Gallente or Caldari space.
If you wanted to be SOE aligned, (or even if you didn't) you could even solo a few wormholes in your cloaky probers and bring your freed slaves to become settlers on new, undiscovered planets.
The SOE is supposed to be a humanitarian organization, but as far as I know they still support slavery and the use of Vitoc.
Of course, your corp could also be a 'front' organization since the Underground railroad slave smuggling organization are considered as terrorists by the Amarr state.
As far as a political machine, well it doesn't get more political than slaves and their freedom.