The FS1 tech description of the GTA implies that the unification of Terran systems was a peaceful and fairly straightforward process, whether or not that's realistic. Some people have made campaigns where the GTA's unification involved some kind of war, though, and that would also be an acceptable route to take.
If you haven't already, check out
this wiki entry on the basic structure of the GTVA federal government. We don't know how much authority the GTVA federal government has over its constituent systems. But given that the GTVA is designed to organize a unified defense against the Shivans, I think we can reasonably presume that it has considerable central authority, at least in anything related to defense. Each system probably has its own democratic or pseudo-democratic government, I'd presume, though individual system governments might vary considerably in structure, policies, and level of authority.. Beta Aquilae is the capital of the GTVA. According to
this list, only Alpha Centauri, Altair, Vasuda Prime, Aldebaran, and Deneb are Vasudan systems.