Despite the League being specifically formed to resist the Hegemony we really know nothing about how they've antagonised the Hegemony other than a suggestion that they've assisted Andrada in maintaining Sindrian independence (we know more about their conflicts with the Luddic Church than the Hegemony). As it is the League feels like a polity like any other instead of a federation of independent states, with most of the lore being about how they've consolidated their power (or Kazeron's power?), and some of the member states don't have any lore about their culture, politics, motivations for joining the League, etc.
From what I can gather, the independent polities of the League are:
Kazeron: a stratocracy, probably not dissimilar to Sindria except without a sole dictator. The few military exploits of the League we know of seem to be more properly Kazeron's exploits, e.g. Mazalot and Isirah. Not counting Mazalot's exiled regime, this is the only League polity that has a permanent presence in more than one system (Laicaille habitat). There's also the independent world of Eldfell in Thule which we know nothing of: given Kazeron's militancy there must be something going on here.
Mazalot: an unstable polity prone to regime changes, in part because of a large Luddic population. Despite being the largest or second largest League population, Mazalot seems more like a client state than an equal partner in the League; with little power even in its own system seeing how Yesod is handled.
Salamanca: I'm surprised this is a League world instead of an Independent or Pirate world - a backwater industrial planet taken over by a corrupt/criminal ex-Mazalot government. I guess the League really doesn't care about domestic politics and will take anyone that will swear fealty. The question is what the Salamancan government gains from re-applying for League membership after their exile: there's an equally large Pirate world in the same system, so it doesn't seem like they're under threat from the Hegemony.
Mairaath: the tragic story of the enlightened brought low by ignorance and mindless hate. I always wanted to know: what was the canonical end to The Last Hurrah? In the end it probably doesn't matter, as even if the Mayasurian fleet won the day the Hegemony will send more, and eventually they will be overcome. I'd like to think that the Mayasurians prevailed that day, and with that unexpected victory were able to secure League membership and spare their broken world from Hegemony eradication.
The Westernesse League: a mini-league consisting of Fikenhild, Athulf and Suddene. Other than Fikenhild being the system representative we know almost nothing about the individual polities. There's also the independent world of Ailmar that again we know nothing about, although there must be some background as to why it isn't part of Westernesse's mini-league.
Madiera: again we know nothing about this polity. Presumably Cibola is ruled by Madiera rather than an independent polity in itself.
Olinadu: we know they joined the League to protect themselves from the Luddics. Abundant volatiles aside, the League probably sees it mostly as a buffer-state against the Church.