Assuming your car can be accurately described by a dambed harmonic oscillator when oscillating undisturbed, the differential equation defining its vertical motion is:

$$
\frac{d^2y}{dt^2}+a\frac{dy}{dt}+by=0
$$

Where $a$ is the damping coefficient and $b$ is the spring constant. It's characteristic equation is:

$$
r^2+ar+b=0
$$

In principal, there are three different cases depending on the nature of the roots of this equation, but assuming your car will exhibitit exponentially decreasing oscillations, we can assume that it has two complex roots, $r_1=\alpha+i\beta$ and $r_2=\alpha-i\beta$ and the general solution to the differential equation will be:

$$
y=e^{\alpha t}\left(C_1\cos\beta t+C_2\sin\beta t\right)
$$

If you initialize oscillations with large amplitude in your car manually and then record $y(t)$ with a camera, you should be able to estimate $\alpha$ (basically by registering how fast the amplitude decrease between each oscillation). Note that $\alpha$ should be negative. The easiest way to do this might be to plot the natural logarithm of the highest point reached for each oscillation against time. You should get a straight line with $\alpha$ as slope. Just remember using the car's rest position as baseline when measuring $y$.

Once you have $\alpha$, just use that:

$$
(r-\alpha-i\beta)(r-\alpha+i\beta)=r^2+ar+b\\
\Rightarrow r^2-2\alpha r+\alpha^2+\beta^2=r^2+ar+b\\
\Rightarrow a=-2\alpha
$$

to calculate your damping coefficient.