They never mention the ducks in the main game, nor do they even hint at the many ridiculous steps required to find them. To start, you need to talk to two random NPCs on opposite sides of the map, in places you have no reason to visit. Once you speak to them, walk down an obscure back alley, and pick up a bronze medal featuring a schoolgirl's face.
Or pick up actual women from elsewhere in the lot. Your choice.
Whoa there, partner! Don't spend that schoolgirl token at the used-panty machine just yet -- you need to use it to unlock the third floor of a local arcade. Inside the arcade is a fight club featuring the two NPCs you spoke to earlier. You must beat the tar out of them to continue and, of course, they're the toughest fighters in the game. Even though one of them is an unassuming shopkeeper. Once you win, you get the silver medal, and the shopkeeper introduces you to duck racing. You can bet on your favorite duck and everything, but no matter how often you win, you won't get the gold medal until you win with your own duck. Where do you get a duck? Why, from the duck tree, of course!
"Oh great, they're still in season!"