You can use Gauss' Law to find the electric field due to a given charge distribution. However, it is very limited because Gauss' Law only deals with the divergence of the electric field. If you convert this to an integral equation, then it relates the total enclosed charge to the electric flux through a Gaussian surface.
If we have a problem that we can capitalize on symmetry, then we can sometimes solve for the electric field. For example, given that we have a point charge in space, we can choose a spherical shell as our Gaussian surface. Then, we know by symmetry that the electric field vectors must be normal to the Gaussian surface. Thus, the flux through the surface at a given point is equal to the sign and magnitude of the electric field vector. From this we can derive the electric field due to a point source. Likewise, we can use this again for any spherically symmetric charge distribution (like a uniform charge density spread across a sphere's surface or volume).
But once we lose these symmetries, then the flux will not be enough to uniquely define the electric field.