There is a multispecies "Federation of Stars" in Edmond Hamilton's Interstellar Patrol series, beginning with his novelette "The Star-Stealers", first published in Weird Tales, February 1929, available at the Internet Archive, and reprinted in Avon Fantasy Reader No. 6, 1948, also available at the Internet Archive. The Interstellar Patrol stories were collected in Hamilton's 1965 collection Crashing Suns.
The following excerpt is from "The Star-Stealers" (emphasis added):
It was the yellow star that I was watching, now, as our ship fled on toward it at eighty times the speed of light; for more than two years had passed since our cruiser had left it, to become a part of that great navy of the Federation of Stars which maintained peace over all the Galaxy. We had gone far with the fleet, in those two years, cruising with it the length and breadth of the Milky Way, patrolling the space-lanes of the Galaxy and helping to crush the occasional pirate ships which appeared to levy toll on the interstellar commerce. And now that an order flashed from the authorities of our own solar system had recalled us home, it was with an unalloyed eagerness that we looked forward to the moment of our return. The stars we had touched at, the peoples of their worlds, these had been friendly enough toward us, as fellow-members of the great Federation, yet for all their hospitality we had been glad enough to leave them. For though we had long ago become accustomed to the alien and unhuman forms of the different stellar races, from the strange brain-men of Algol to the birdlike people of Sirius, their worlds were not human worlds, not the familiar eight little planets which swung around our own sun, and toward which we were speeding homeward now.
The headquarters of the Federation, on a planet of Canopus, are described are described in this excerpt from the next story in the series, "Within the Nebula", first published in Weird Tales, May 1929, also available at the Internet Archive.
Down we sped toward the mighty structure's base, down over the great buildings on either side which housed the different departments of the Galaxy's government, down until our ship had come smoothly to rest on the ground a hundred feet from the tower itself. Then the ship's hull-door was clanging open, and a moment later I had stepped onto the ground outside and was striding across the smooth sward toward the mighty tower. Through its high-arched doorway I passed, and down the tremendous corridor inside toward the huge doors at the end, which automatically slid smoothly sidewise as I approached. The next moment I had passed through them and stood in the Hall of the Council itself.
Involuntarily, as always, I paused on entering, so breathtaking was the immensity of the place. A single vast circular room, with a diameter of near two thousand feet, it covered almost all the mighty tower's first floor. From the edge of the great circle the room's floor sloped gently down toward its center, like a vast shallow bowl, and at the center stood the small black platform of the Council Chief. Out from that platform back clear to the great room's towering walls were ranged the countless rows of seats, just filling now with the great Council's thousands of members.
Beings there were among those thousands from every peopled sun in all the Galaxy's hosts, drawn here like myself each to represent his star in this great Council which ruled our universe. Creatures there were utterly weird and alien in appearance, natives of the whirling worlds of the Galaxy's farthest stars—creatures from Aldebaran, turtle-men of the amphibian races of that star; fur-covered and slow-moving beings from the planets of dying Betelgeuse; great octopus-creatures from mighty Vega; invertebrate insect-men from the races of Procyon; strange, dark-winged bat-folk from the weird worlds of Deneb; these, and a thousand others, were gathered in that vast assemblage, forms utterly different from each other physically, but able to mix and understand each other on the common plane of intelligence.