Distribution of charge within the region where the field is located, is obviously uniquely defined, because it's just
$$\rho=\epsilon_0\nabla \vec{E}$$
However, if you cut out a region of space, and want to predict the contents of this region based only on the field outside, you can't do it in a unique way.
The reason is that you have too many degrees of freedom. If you decompose the field outside into the inhomogeneous contribution of the charges outside, plus the extra that was supposed to be caused by the cut out region (and obeys the Laplace equation outside the region), then the second contribution (the one with $\nabla \vec{E}=0$ outside the region) can be exactly reproduced simply by putting the correct charges on the surface of the boundary! That's what many call the holographic principle. You see now that there is one unique surface solution for every bulk distribution - but however differently you choose to distribute the charges inside the region, you can always rearrange the surface charge to accomodate the same external field. That's how conductive objects "mask" whatever you put inside them.
So... charges that occupy the space where the field is known, are uniquely defined. Charges in regions where the field is unknown, are arbitrary. Charges on the boundary of the known region are uniquely defined for each particular choice of distribution inside - assuming no charge is inside, and all field is caused by the surface, is simply one of the infinite number of solutions.