I'm going to give an opposing answer.
As long as the mass is spherically symmetric, you shouldn't be able to determine its distribution, because spherically symmetric changes in the mass distribution shouldn't change the resulting gravitational field.
If the Sun, the planets, and the other bodies in the solar system truly were spherically symmetric, and that could be determined about those objects by observing how objects orbit another would be their masses. (More precisely, what would be determined are the standard gravitational parameters $\mu=GM$, the product of the universal gravitational constant and mass. The large uncertainty in $G$ means that mass can only be determined to four decimal places of accuracy.)
Masses in the solar system aren't spherically symmetric. The Sun is the closest thanks to its large size and its slow rotation rate. The giant planets have a marked spherical bulge, but also apparently have some gravity anomalies due to uneven mass distribution. The terrestrial planets are even less spherical than are the giant planets, the asteroids, even less so. It's best to describe asteroids and moons with a radius smaller than 200 to 300 kilometers as lumpy potatoes rather than spherical cows. Vesta is right in the middle of that range. While not near as lumpy as 25143 Itokawa (visited by JAXA's Hayabusa a decade ago), Vesta is still far from spherical. This gives a handle for measuring characteristics within Vesta, just by observing how Dawn orbits Vesta.
Numerical orbit determination and propagation techniques account for the non-spherical nature of solar system bodies and for perturbing effects from other bodies. The Earth-based tracking stations used to communicate with Dawn provided measurements that give clues regarding how Dawn was orbiting Vesta, which in turn gives clues to Vesta's gravitational field. The most precise measurement is the Doppler shift in signals sent from Earth to Dawn which Dawn simply relays back to Earth. (Every deep space probe launched by the US and Europe is outfitted with a device for just that purpose.) Enough of these observations, spread out over time, coupled with models of the solar system, models of how gravity works, plus lots of computing horsepower, gives a picture of the interior of a solar system body.
One way to model the gravitational field of not-quite spherical objects is with spherical harmonics (or sometimes elliptical harmonics). The mathematics is very well established. This approach works nicely for planets, somewhat nicely for Moon-sized object (but the Moon has five large mascons (mass concentrations) on the near side and a two kilometer offset between its center of mass and geometrical center). They might work even for a Vesta-sized object.
Another approach is to use the body's geometrical shape to model the gravitational field. Dawn took lots of pictures of Vesta. These pictures collectively yield a 3D shape model of Vesta. A commonly used approach in computer graphics is to represent the surface of some object via polygons (typically triangles). These shape models can be used as a gravitational model by projected those polygons to some well-chosen center to create polyhedra. The gravitational field for a constant density polyhedron is easy to calculate (with a computer, that is), and from that, you have a polyhedral gravity model of the object in question.
The orbits from the harmonics-based and polyhedra-based gravity models inevitably won't agree with one another, or with the observations. The biggest problem is almost certain to be the assumption of a constant density in the polyhedral gravity model. A Vesta-sized body should have some amount of differentiation. Polyhedral gravity models can be augmented with a density profile. Modifying that model so it is more consistent results with the observations gives a clue as to what's inside Vesta.
Alex S. Konopliv, et al., "The Dawn gravity investigation at Vesta and Ceres," The Dawn Mission to Minor Planets 4 Vesta and 1 Ceres, Springer New York, 2012. 461-486. (Same reference used by Hohmannfan)
Robert A. Werner and Daniel J. Scheeres, "Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia," Celestial Mechanics and Dynamical Astronomy 65.3 (1996): 313-344.