Some players report that they are unable to fire their gun while seated in the car during the Hillcrest car chase sequence.
This issue occurs when the Melee action is bound to the same key as Shooting. For example, when both actions are set to Left Click.
As a workaround, change your controls so that Melee and Shooting use different keys or reset your controls to the default layout. Once the conflicting keybind is removed, you will be able to shoot during the Hillcrest car chase.