For an infinite plane, I know that that using Gauss' Law, the field is simply $σ/2e_0$. However, this is true because the net field is directly upward everywhere. If the plane isn't infinite, this wouldn't be the case, but the net field at the very center would still be directly upward. Is there a way to use this symmetry to simplify the problem or would I have to integrate both over the length and the width keeping in mind the i,j,k components of the field?
Yes, you can use that symmetry so that you only need to find the component of the field perpendicular to the plane.$^*$ However, you will still have to use Coulomb's law to find the field here, so you will still need to perform integration.
$^*$You can calculate the other components of the field if you want, but you will find them to be $0$.