No, they didn't. In Star Wars Episode II: Attack of the Clones, the Jedi Knight Obi-Wan Kenobi travels to the secretive planet of Kamino, where he discovers the existence of a clone army that would feed the so-called Clone Wars and eventually serve as the model for the evil Galactic Empire’s stormtrooper infantry. (Spoiler alert: Don’t watch the prequels.) Those clones weren’t white in any sense of the word. Jango Fett, the bounty hunter who served as the genetic template, was culturally (and perhaps ethnically) a Mandalorian. And the actor who portrays him, Temuera Derek Morrison, is a New Zealand-born person of brown skin and partial Maori descent.
Even if Morrison and Fett (and all of his clones) choose to pass as white, by the time of the events of Episode IV: A New Hope, the Empire has been recruiting from general populations for years. That’s why it makes sense that a young Luke Skywalker, lured by a galaxy larger than the humble moisture farm he calls home on Tatooine, dreams of enlisting in the Imperial Navy.
The Empire is not a racially diverse institution. Everyone we meet in the Imperial officer corps, for example—the chain of commanders that Darth Vader is always Force-choking out—is white (and seemingly British). That said, if the Empire recruited black stormtroopers, that would be totally in keeping with a racist Empire—since black and white people are not actually from different races, the way that (say) humans and Wookies are.
What distinguishes the Star Wars universe from the Star Trek universe is the fact that, outside of the goose-stepping ranks of the Imperial military structure, its universe is pretty chill on race, if not totally post-racial. No one in the Rebellion bats an eye when Lando Calrissian and Nien Nunb copilot the Millennium Falcon during the Battle of Endor in The Return of the Jedi, even though the former is a black man from Bespin’s Cloud City and the latter a fish-faced smuggler from the humid caves of Sollust. Neither of them appears to mind taking orders from Admiral Ackbar, the high-ranking Mon Calamari military mastermind, either. Han Solo even learns a lesson about doubting the diminutive Ewoks, whom he initially regards as primitive.
The most glaring instance of everyday prejudice across the entire original trilogy takes place in the notorious Mos Eisley cantina. “You will never find a more wretched hive of scum and villainy,” Obi-Wan tells Skywalker as they venture in. A sketchy place, maybe, but not because Ithorians and Rodians drink alongside Aqualish and Corellians. When the bartender says, “We don’t serve your kind here,” he’s talking to the droids—marking him as a Luddite, perhaps, but not a racist, exactly.
The Star Wars films themselves commit racism, specifically the prequels, which seem to mock Jews and Asians with their depiction of two alien races (galactic bankers and traders, of course). Which is to say nothing of Jar Jar Binks, a walking crime against black Caribbeans. And the expanded universe depicts racism: The Emperor disdains non-human races, for example, in the comics and novels that flesh out his back story (which are not considered canon).