The main source of missions in the Tundra Express is Tiny Tina, the crazy bomb maker. You will meet her during the course of the main Story and her side missions involve a crazy tea party that she's trying to set up. The first two missions involve getting the party guests, supplies and the guest of honor while the last mission of the three is a defensive mission where you have to protect the tea party from bandits.
There is also a mission in the Tundra that is started by killing the Tundra Patrol. Killing him will set you on a mission to gain his treasure.