In career mode, you will occasionally receive opportunities from sponsors. Unlike previous Nascar games, these sponsors expire and give you fixed payouts. When a sponsor expires, you can choose to renew it. Once you have ten different sponsors stamped on your car, the achievement will unlock.
Note: You have to manually stamp your sponsors on your car in the paint booth. Scroll over to decals, create a new layer, and stamp it wherever you like. I do not actually know what happens when you don't (the game says "sponsors like to be seen on your car" or something, so if you can add to my solution that'd be helpful!
Achievement unlocks once the 10th unique sponsor has been acquired.