In all honesty, I imagine that the Fleet started with the Standard. From there, I can imagine that a few Rifters who went "Well, the guy running the Standard is a reasonable fellow and his associated queendom was not particularly hostile, so..." and joining up for mutual profit. Missions happen. League wolfpack notices their success and signs on, bringing up the civilian count and fleet strength.
At about the same time, a Kai Queen and its associated Drones that partook in the Technokai Dominion contribution to Grand Fleet decide that their interest begins and ends with their own amusement, splits from the Dominion, and thinks "Okay, so what are ways we can get paid to do what we want? Oh, join a mercenary fleet. Well, there's this one that is made of Useful Not Kai and Not Kai; let's see if they're willing to deal." Enough people remember the Queen to go "Oh, okay, she's part of the almost trustworthy group, I remember not being killed by that ship when it could have, plus profits, less chance of dying, eh why not, it's better than leaving it to go around terrorizing planets and our next encounter being a shooting encounter."
Hearing about a mercenary group that is possibly being manipulated by the League and Kai, the Vanguard decides that this group warrants watching, and joins up. The Humans go "Alliance scum, but at least they can stop the Kai if they do what we expect them to someday"; the Rifters go "Well, we stand to make money and they're mostly good at keeping their word", and so on down the line. After enough missions, they are tolerated on the grounds of "Yeah, they're hideous wasp aliens who have no morals or scruples beyond self interest, but they're OUR terror wasps."
Or "We're mostly opposed to slavery, but we are even more opposed to stupid pointless deaths, so we'll just kinda tolerate the Kai enslaving our enemies."
They're the Jayne, if I want to continue the Firefly references. *shrugs*