This work investigates the anomalous transport of passive tracers in a three-dimensional, quenched velocity field generated by static vortex filaments, a system theoretically predicted to exhibit superdiffusion governed by Lévy-stable Holtsmark statistics. Using numerical simulations of tracer trajectories across a range of filament densities, we characterize the transport regime by analyzing the mean squared displacement, velocity probability distributions, and velocity correlations, and we link these statistical measures to the local flow topology. Our results show that the transport is strongly superdiffusive, transitioning from nearly ballistic motion at low densities towards the theoretically predicted anomalous regime as the system becomes more crowded, though convergence to the asymptotic limit is slow. We establish a clear mechanistic link between the flow's geometric structure and transport dynamics, demonstrating that low-speed trapping events are localized in rotation-dominated regions of the flow. Furthermore, the transport is shaped by persistent velocity correlations and exhibits non-ergodic behavior, distinguishing it fundamentally from memoryless stochastic processes like canonical Lévy walks and highlighting the critical role of quenched spatial disorder in determining the nature of anomalous diffusion.