You can make reservations for Big River by calling them directly. You cannot make them through Disney Dining.
Anyway....please don't waste your time and money. We had lunch there in June on our last day at WDW. The food was not bad, but I thought very overpriced since it's not a Disney restaurant. The inside had some kind of musty odor and was not up to Disney standards in any way. Since there are only single doors to the outside the flies were able to get inside very easily and made it very annoying while we were eating.
There are so many better places to eat in the boardwalk area. Go over to allearnet.com and click on dining and menus for the Epcot resorts (Boardwalk, Yacht Club, Beach Club, Swan and Dolphin).