No, that isn't accurate. It's been explicitely stated since Revenge of the Sith came out that the clone troopers of the GAR became the first stormtroopers (Here), and that they were gradually replaced by clones of other individuals (such as Erv Lekauf, although he's not a great example as he was an Imperial Army officer, not a stormtrooper, he was cloned only six times, and none of his clones actually became stormtroopers--I can't find a better specific individual) and by human recruits (such as Davin Felth) as the Fett clones died of old age (accellerated aging doesn't stop later in life), so that by around 9 BBY there weren't many Fett clones left in service. Wookieepedia says that
As the war dragged on, especially after the Battle of Endor in 4 ABY and the destruction of the primary stormtrooper academy on Carida in 11 ABY, the Empire began running short of properly trained and equipped stormtroopers and turned to lower and lower quality substitutes. Eventually this resulted in the Empire of the Hand and Imperial Remant allowing for nonhuman stormtroopers, which looks like this. So in fact, while we had a pretty good handle on what stormtroopers were before the reorganization, given that The Force Awakens is set well into what would have been the Imperial Remnant era in the old chronology, the field's still pretty much open right now. We haven't seen any stormtroopers that aren't wearing human armor in the trailers, but there could easily be, for a random example, Chiss or Neimoidians under those helmets.