Electric field due to a uniformly charged FINITE rectangular plate

The integrals are difficult but not impossible, unless I've made a mistake with WolframAlpha. The result is:

$$E = \frac{\sigma}{\pi \epsilon_0} \arctan\left( \frac{ab}{4r\sqrt{(a/2)^2+(b/2)^2+r^2}} \right)$$

When $a,b \to \infty$ the whole arctangent goes to $\pi/2$ and we recover $E=\frac{\sigma}{2\epsilon_0}$, which is definitely encouraging.

And I don't know what you mean by "directly solving Poisson's equation". As far as I know, the usual way to do that is to use Green's functions, i.e., this integral.