The Big Bronze Medal is an award that can be obtained by Ryo Hazuki. Once this medal is obtained, Ryo has access to the street fights in Pine Game Arcade (thus it, along with the Big Silver Medal, are necessary to unlock The Duck Races).
The most simple way is to grab it from the Vacant Lot in the Green Market Quarter. Another way is to give the waitress at the Lai Lai Eatery in Aberdeen some money after causing trouble while chasing Wong. The last way is to win the Roll It On Top stand in the Lucky Charm Quarter five times in a row.