We propose a stochastic model for molecular transport at the nanoscale that describes the motion of two-dimensional molecular assemblies called multivalent random walkers (MVRWs). This walker model is an abstract description of the motion of multipedal molecular assemblies, called molecular spiders, which use deoxyribozyme legs to move over a surface covered with substrate DNA molecules, cleaving them to produce shorter product DNA molecules as they go. In this model a walker has a rigid inert body and several flexible enzymatic legs. A walker moves over a surface of fixed chemical sites. Each site has one of several molecular species displayed, and walker legs can bind to and unbind from these sites to move over the surface. Additionally, the enzymatic activity of the legs allows them to catalyze irreversible chemical changes to the sites, thereby permanently modifying the state of the surface. We describe a MVRW system as a continuous-time Markov process, where all state transitions in the process correspond to chemical reactions of the legs with the sites.We model the kinetics of the leg reactions by considering the constrained diffusion of the walker body and unattached leg. Through kinetic Monte Carlo simulations, we show that the irreversibility of the enzymatic action of the legs can bias the motion of walkers and cause them to move superdiffusively over significant distances.