Motzkin spin chains and their area-weighted deformations are a countinuous family of one-dimensional frustration-free Hamiltonians, whose ground states exhibit a novel quantum phase transition. By tuning a single parameter, they go from a phase obeying an area law to a highly entangled rainbow phase, where the half-chain entropy scales with the volume. Using the representation of these ground states as superpositions of random walks, we introduce tensor networks for these ground states where local and global rules of the walker are baked into bulk tensors, thereby providing an efficient description of the ground states (some of which satisfy a volume law scaling of entanglement entropy).