You can wait for a car challenge to emerge in the main screen's posters; if you place first, you'll win the automobile. Car challenges are free to try once, but if you don't place first, you'll have to pay gems to try again. Car difficulties arise after the player reaches Level 3.
You can win them by participating in Daily Challenges. Daily Challenges come when a player reaches the fourth level. If you complete the challenge, you will be given the opportunity to spin the reward wheel. You can win the car if there is a yellow ribbon on the steering wheel. The first spin of the wheel is free; subsequent spins cost jewels.
You can buy them outright for gems at the Team screen.