You earn the ability to do Wasteland missions after you complete the story mission Into Madness
Once you complete that and are in Jeet's Stronghold you can accept a new missions from Chum, as well as a Wasteland Missions, being Dinki-di
To complete the mission simply fast travel to the vantage outpost closest to Chum's Hideout and drive the rest of the way. Once in put the dog in the back of the buggy and hop in.
Drive the buggy back to Jeet's and hold
to complete the project next to the car.
Go through the prompts and Voila!