In World of Warcraft:Legion and Warcraft III: The Frozen Throne, Naga play a significant part of the Illidari campaign, however, most of the naga seen are servants of the old gods. Are all naga servants, or were some freed from N'Zoth's control?
The only Naga seen to act separately serve the Illidari.
The Coilskar Naga that were part of the tribes that originally responded to Illidan's request for assistance in Warcraft III: The Frozen Throne, and currently serve as part of the Illidari demon hunters support structure, are the only known independent clan of naga... and their independence may have varied and can be debated.
Naga are somewhat unique in that by and large, almost all of them revere their Queen Azshara. There are a lot of naga that just randomly show up on shoreline to raid, pillage, and enslave, and it's hard to confirm if all of their motivations are the same when World of Warcraft uses them liberally as ocean and coastline bad guys. Other races have been used as frequent bad guys and had more complicated motivations - the Dark Irons in theory all served their Emperor and the command of Ragnaros the Firelord, but the Thorium Brotherhood broke away during that time, and now with Ragnaros defeated and Moira Bronzebeard leading the clan, Dark Irons are part of the Alliance. Still, there aren't any examples at all, that I can find, of any of the Naga speaking against their Queen. Since Queen Azshara made the pact that created the naga with the Old Gods and serves them (more specifically, the Old God N'zoth) ever since, it can be assumed with widespread devotion the naga follow their Queen and thus the Old Gods.
Then we have the Coilskar. Lady Vashj initially led this tribe, one of many tribes she led under Illidan, and appeared to do so at Queen Azshara's behest. Vashj is named as one of Azshara's original handmaidens so her ultimate loyalties seem quite clear. With Vashj dead as of The Burning Crusade, the Coilskar were forgotten by most until Legion, when the demon hunters known as the Illidari re-established themselves and called upon the Coilskar as allies. So far they appear to be doing so faithfully. It is still possible however, that serving the Illidari serves the purposes of the Old Gods. The demons of the Burning Legion intended to destroy Azeroth; the Old Gods wouldn't have had a planet, a world-soul, to continue corrupting and turn to their design if the Burning Legion had succeeded.