Most likely, the transistor is being used as the switch.

Transistors have three terminals; 1) Base, 2) Collector, and 3)
emitter.  In this application, I'm going to guess that the power is
tied to the collector and that when the transistor is forward biased,
voltage and amperage flows from collector to emitter thereby providing
power to the motor to run.

My suggestion is that you take the circuit board to an electronics
repair shop.  If my suspicions are correct about the electrical flow,
just about any radio or TV shop should be able to make a reliable
repair.

The thing you should be worried about is WHY the circuit became
overloaded.  Transistors are rated on the maximum amount of amperage
capable of flowing through the transistor from collector to emitter.
If the original transistor just wasn't able to withstand the normal
amperage during the life of the vehicle, no problem; just replace the
transistor with one able to withstand a larger amperage flow.

So, the first thing would be to measure the amperage flowing in the
circuit to make a determination if the original transistor was of the
correct size to handle the current.  Take the measured amperage
information with you when you transport the circuit board to your
chosen repair shop.  Make sure you take note of the "start-up"
amperage and not just the "running" amperage.

Transistors are manufactured in NPN and PNP configurations.  NPN is
the most common type.  You must replace the defective part with the
same type or other steps must be taken to ensure that the resistors
which bias the transistor are of the correct size and are placed to
cause the transistor to forward bias properly.

Transistors have the ability to use a DC voltage to carry an AC
signal.  Most likely, the circuit board toes not generate an AC
signal, but that cannot be ruled out until it is looked at by a
qualified technician.  Another component that is likely present on the
circuit board is either a 555 or 556 solid state timer used to control
the amount of time the transistor is switched on..

Unless you have the correct level of training to replace the
transistor yourself, I would strongly suggest that you find a
qualified technician.  Transistors may be biased using several
different techniques depending on the application.  I suspect that the
transistor in this particular application is "based biased".  If you
do not understand these terms as they are applied to transistors, do
not attempt to make the repair yourself!

A discrete electronic tutorial is beyond the scope of this medium.
Please do not ask for further explanations for any of the
aforementioned terms.  The summary above is a two year culmination of
both analog and solid state electronics training including many
mathematical formulas needed to properly bias transistors in single
and multiple stage amplifiers.