In Legends the X-Wing was originally designed for the Empire, Until Incom defected with the plans.
That is why it is primarily Rebellion (and offshoots) based and basically the Symbol of the Rebels.
As for non-Rebellion usage.
The Lightspeed Panthers (per Star Wars: The Essential Guide to Warfare) were a volunteer starfighter squadron that assisted the Rebellion but were not directly part of it and they flew X-Wings.
In X-Wing: The Krytos Trap, the pro-Imperial Palpatine Counterinsurgency Front has a squadron of X-Wings that were dressed up to resemble Rogue Squadron and used to try to impersonate them; unfortunately (for them) they ended up under the guns of Warlord Zsinj, who was happy to destroy them out of hand either way. It's not stated where the PCF obtained the X-Wings.
In X-Wing: The Bacta War, Ysanne Isard's fighter commander, Erisi Dlarit, tries to sell Isard on the merits of switching at least some of their squadrons from TIEs to X-Wings, which she considers to be superior fighters. She's overruled largely on the basis of Imperial nationalism; Isard thinks of the Alliance (including its fighters) as inferior and what she says, goes. Again the potential source isn't specified, but it may be worth noting that, lacking a factory for them, Isard's forces had been buying TIEs on the gray market.
Post-Galactic War X-Wing varients were used by many organizations in the galaxy that used them that were not exactly directly linked e.g. New Jedi Order and RASP.
Jedi Squadrons were formed mainly during the Yuuzhan Vong Crisis and seen in The Swarm War and through the Legacy of the Force Series and Fate of the Jedi series where they were still flying StealthX's.
RASP (Reconstruction Authority Space Patrol, aka Reconstruction Police) is a policing force that was setup by the Galactic Federation of Free Alliances post Yuuzhan Vong Crisis, mainly seen during The Swarm War and onward.
In canon, The Partisans (Saw Gerera's followers, as seen in Rogue One in the Jeddah scenes) used X-Wings also, and they were not included as part of the Rebel Alliance so that kind of fits.
It's also stated in canon that the X-Wing was is general rotation before becoming the almost exclusive fighter for the Rebellion.
Saponza from the game Star Wars: Commander was a mercenary that flew X-Wings but then joined the Rebellion. Not sure if that is pure canon.