According to the book, Lucy had two affairs in college that didn't do anything for her. Her second lover complained about her being "too big down there." She was Connie's best friend. While hanging around Connie, she heard Sonny's wife describing his size. Lucy got the idea that Sonny could accommodate her--that he would be the one man who wouldn't be put off by her own size. And the affair was convenient while she was hanging around Connie.
As for Sonny: He was probably attracted by Lucy coming on to him--it'd appeal to his machismo. The novel doesn't say that a) Lucy was fat; b) that she was his exclusive extra-marital squeeze; or c) that he made it with her after the wedding, although we might infer that from her yearning after Sonny after his death. The NYC liaison with Lucy was a creation of the film, and merely served to make a transition to Sonny's beating of Carlo.