Usually batteries are used for this, as they can store far more energy and if they're big enough, their maximum power is also large enough for quick charge/discharge. This means that an ultracapacitor isn't necessary for power density reasons and the battery is better for energy density reasons.

For example, a number of electric cars are already traction-limited for acceleration and capable of driving two to three times any finite speed limit they encounter, while still having a 300-400 mile range at full charge.

Conversely, late-generation Honda Insights use a small, but high efficiency motor that is used mostly to charge a battery. It turns on, runs at max efficiency charging the battery, then turns off when the battery has enough charge. The battery provides electric traction and is capable of both load-leveling (since the motor is providing more power than is usually necessary for driving at any given moment) and is capable of providing more power than the motor is in short bursts! This drivetrain is not entirely dissimilar from a diesel-electric train.

Due to ultracapacitor's low energy density, they are only good where you need extremely high power densities (higher than are generally needed by modern electric vehicles) or extremely high charge/discharge cycles. So they've been examined for public transit purposes (where they might not even be carried by the vehicle itself, but rather be installed in a fixed location and connected to the vehicle by rail or overhead lines), but they don't really make sense for cars and trucks.