No. As in the novel, Fats is not an supernaturally animate doll, but the inanimate dummy of a seriously disturbed ventriloquist who believes Fats is alive. Fats is actually an outlet for Corky's insanity, his alter-ego who says the things he cannot.
Then again, there are moments in the movie that make you wonder, various borderline moments designed to keep the possibility in the audience's mind, like Fats moving his head and blinking his eyes when Corky leaves the couch shortly after being rejected by Peggy.
