A novel iterative approach is presented to estimate Young's modulus in homogeneous soft tissues using vibration sonoelastography. A low-frequency (below 100 Hz) external vibration is applied and three or more consecutive frames of B-scan image data are recorded. The internal vibrational motion of the soft tissue structures is calculated from 2D displacements between pairs of consecutive frames, which are estimated using a mesh-based speckle tracking method. An iterative forward finite element approach has been developed to reconstruct Young's modulus from the measured vibrational motion. This is accomplished by subdividing the 2D image domain into sample blocks in which Young's modulus is assumed to be constant. Because the finite element equations are internally consistent, boundary values other than displacement are not required. The sensitivity of the results to Poisson's ratio and the damping coefficient (viscosity) is investigated. The approach is verified using simulated displacement data and using data from tissue-mimicking phantoms.