This may sound sad, but I've always thought like woody was bought in a shop and given to Andy as a Christmas present from his dad. Because Woody never mentioned his dad, and that he looked brand new when Andy got him. But that is my opinion.
A harsh question . I mean, like all you guys said , he 's supposed to be "an old family toy ", probably belonged at Andy's father . It's pretty moving, but I really wonder if it can be true . I mean, think on this way : through all the trilogy, Woody experiences things he was not used to : being replaced by another toy in Toy Story 1, realizing that children grow up and eventually leave their toys behind in Toy Story 2, and of course in Toy Story 3, the moment of Andy growing up is finally arrived . All those things, Woody seems to experience for the first time, while IF he had been Andy's father toy, he WOULD know all those things and would not really worry about, or at least he could tell the others toys what he's been through . (Okay, someone already said this , in simpler words, but heywhatever, I'm still posting it !)
I might have a plausible theory. We know that, save for a few exceptions, toys stay in a sort of stasis mode until taken out of the box. So what if Woody was bought, yet never opened for some reason, and was kept in a garage for years until there was a yard sale and in came mom with a toddler Andy, who fell in love with the toy and mom bought it for him.
Because I know some will point out that Stinky Pete was awake inside of his box, I think that, after a time, and when in constant movement, a few toys might wake up and be conscious of things around them, but it is a rare occurrence, and besides, if Woody was kept undisturbed in a dark and isolated place for years, there would be no reason for him to wake up.
Andy's mom said that Woody is "an old family toy". John Lasseter said that he and the crew always felt that Woody was a hand-me-down toy from Andy's father. Andy's mom refused to sell it b/c she knew that Woody was Andy's favorite. Woody was one of the last things that Andy's father gave to Andy before he died.