Vehicles are unlocked by a number of different objectives, to view what you need to do to unlock certain vehicles just select an event in career and you'll be brought to the vehicle select screen. You can select any of the locked vehicles and the description of how to unlock it will be shown.
Completing career should unlock every vehicle except the Tardan Senior Trophy Truck, for which you need to gain 100 opponent knock-offs. To be able to get opponent knock-offs quickly you can go to Circuit #7 once you unlock it and select the race that pits you, in a Class 1 car, against 7 riders on dirt bikes. So If you require a quick fix to gain the knock-offs that's your place. Just drive and ram as many of the dirt bike riders you can and finish the race.