How do you calculate the lift coefficient of an airfoil at zero angle of attack?
The solution and the approximate solution to zero-lift angle of attack from thin airfoil theory can be found in ESDU 98011. The derivation of an analytical solution can also be found in Anderson, Fundamentals of Aerodynamics. $C_{l_0}$ can then be found by multipling it with $2\pi$.

Edited to add classical thin-airfoil result:

$$C_{l_0}=2\int_0^\pi{\frac{dz}{dx}(cos\theta-1)d\theta}$$
$$x=\frac{1}{2}(1-cos\theta)$$

$x$ is the chord line coordinate from 0 to 1; $z$ is the mean camber line height at coordinate $x$, normalized to the chord length; $\frac{dz}{dx}$ is the curvature of the camber line.

Now you just have to numerically integrate to get the lift at zero incidence.