A level set based shape evolution approach is presented for the inversion of elastic waveform data with a special emphasis on the application to seismic full waveform inversion. A conservative representation of the linear elastic equation of motion in 2D is formulated as a symmetric hyperbolic system. The minimization of a suitably chosen least-squares data misfit functional is then performed by a shape evolution approach. This evolution is driven by Kaczmarz type gradient-based descent directions which are practically obtained by solving a time-reversed form of the same conservative elastic system. A level set method is employed for the computational description of the evolving shapes. Different regularization schemes are tested and compared for stabilizing this shape evolution, combined with an additional integrated optimization loop for simultaneously estimating different internal elastic parameter values. Numerical experiments in 2D are presented which demonstrate the performance of this novel shape reconstruction scheme in the application of salt dome reconstruction in seismic full waveform inversion.