The phenomenon of ratcheting, which allows a preferential direction of motion despite unbiased forcing, is numerically studied for matter-wave solitons in a Bose-Einstein condensate (BEC) under the influence of time-dependent, spatially symmetric and periodic, potentials. We consider a potential, which is horizontally vibrated by a periodic forcing. We study the dynamics of the ensuing solitons for single and bi-harmonic drivers of the potential vibration. The study is based on the dynamical reduction of the original model, the so-called Gross-Pitaevskii equation, a partial differential equation (PDE), to a set of coupled ordinary differential equations (ODEs) for the soliton parameters. These ODEs are obtained via conserved quantities and through variational approximations for, respectively, time-independent and time-dependent external drivers. For regular orbits, we use direct comparison between the PDE and ODE dynamics in order to validate our dynamical reduction. For chaotic regions, where one-to-one comparison between orbits is not possible, we use the Lyapunov exponents as a measure for comparison and validation of our reduced model. Finally, we show that by breaking the time-reversal symmetry of the potential, non-zero ratcheting is observed. The optimum ratcheting parameters are found to be in agreement with the general theory of ratcheting in bi-harmonically forced systems.