How does "Lawful Evil" work
The Cardassians, Imperium of Man (most of the time), Galactic Empire, Gamilans, and The Enclave are some examples. Lawful Evil types don't do "evil" acts in the name of mustache-twirling villainy, but to further a goal that may well be noble in its own right. A lawful evil character would be one who is consistent in their actions, follows SOME sort of moral code, and may even be trustworthy and honorable.
tldr, "Lawful Evil" means the entity in question has what could be called "good intentions" behind their atrocities. They aren't evil for the sake of evil, or totally mad like chaotic types.
vigilante, from what i assume
No, vigilantes are typically Chaotic Good. They're going against the
law to fight for
justice.
I think lawful evil is corrupt politician or something
That would fall more within the lines of neutral evil. They're only looking out for themselves.