Can the equation $frac{1}{3}ML^2 $ be used to calculate the moment of inertia in a non-uniform rod?

No. When deriving this equation from the moment of inertia definition, the density $rho$ is assumed to be uniform. If the density is not uniform, the $dfrac{1}{3}ML^2$ equation does not hold.

You would have to use $$I = int r^2 dm$$ equation to find the moment of inertia of your non uniformly dense rod.

EDIT: as noted by user @Andrew Steane, for some 'special' density distributions it would hold.

But as a general rule, no (except in certain cases).

No you cannot use this formula. Imagine that all mass of the rod is concentrated at one end, the the moment of inertia will be either $0$ or $M L^2$. The correct formula for the moment of inertia in your case is begin{equation} I = int_0^L dx rho(x) x^2, end{equation} where $rho(x)$ is the linear mass density of the rod. For a uniform rod $rho(x)=frac{M}{L}$, and then integration leads to $I = frac{1}{3}ML^2$.