Oregon Trail Card Game: What happens when you run out of supply cards?
After reaching out to Pressman Toy regarding the ruling I received the following:


  In the Oregon Trail narrative, players throw supplies overboard to help cross the river. But, even though the wagon may become empty and technically it would be easier to float, they are stuck. Players must continue to follow the instructions until someone finally does roll an even number!


So unless you are dealing with a river where players die when rolling a one, nothing bad happens, but you do have to spend a little time rolling.