i imagine the problem is finding an area of space that's smooth enough to make the measurement accurately. the distribution of mass within would need to be very uniform and "average" (mass curves space; what you're trying to find here is the average curvature). so it would need to be a very large area (since matter in the universe is clumped into galaxies and the like - you want a region so large that those clumps are averaged out).
since the voyagers are only separated by a scale similar to the size of the solar system, and we're talking of sizes much larger than the separation of galaxies (or superclusters), the answer is no (unfortunately) (and of course, at those scales, you have the practical problem of light having a finite speed, so your experiment would take a long time!).
in fact, that's why this comes from cmb (cosmic microwave background) measurements. because those give you, in effect, measurements across the largest chunk of universe observable.
having said all that, measuring changes in curvature is an easier problem. and that's what gravitational wave detectors try to do. and i am pretty sure there have been proposals to do that with satellites (measuring (changes in) distances to each other with lasers).