Can the properties of a volume of space-time vacuum be determined by measurements just on its surface?

It is a generic feature of differential equations in second order partial derivatives that knowing the values of fields on a closed boundary establishes uniquely the solution in the bulk region. Some caveats are:

1. Some equations are degenerate and admit multiple bulk solutions, hence invalidating the claim above. In physics, such situations are usually treated in the following manner: the different bulk solutions are interpreted as the same physical configuration; and they must be related by gauge transformations. Therefore, the validity of the claim is restored when we take gauge symmetry in account.
2. For regions with nontrivial topology, it is possible to have a finite number of degrees of freedom that aren't fixed by boundary values. For the case of a sphere, this doesn't happen, so we won't bother with this caveat.

This is why what you said is true for electrostatics. When the fields don't change in time, the mathematical problem we're solving is a differential equation in second order partial derivatives in 3 dimensions. Hence, values on a 2d spherical boundary are enough to reconstruct the solution, modulo the caveats above.

For fields that change in time, this simplification differential equation in second order partial derivatives in 4 dimensions, and we need to fix the values on a 3-dimensional boundary with a topology of the 3-sphere.

General Relativity can be brought into a similar form (differential equation in second order partial derivatives in 4 dimensions), see e.g. the ADM formalism for reference.

Hence, the same results are true in General Relativity.