Chemical energy is stored in the bonds between atoms and molecules. When these bonds are broken, the energy is released. In a car engine, the gasoline is burned, which breaks the bonds between the carbon and hydrogen atoms in the gasoline. This releases the chemical energy stored in the gasoline, which is then used to move the car's pistons and crankshaft.
The crankshaft converts the reciprocating motion of the pistons into rotary motion, which is then used to power the car's wheels. The wheels roll along the ground, and the car moves forward.
So, while a car does not produce chemical energy, it does use chemical energy to produce mechanical energy.