How the cosmic web feeds haloes, and fuels galaxy formation is an open question with wide implications. This study explores the mass assembly in the Local Group (LG) within the context of the local cosmography by employing simulations whose initial conditions have been constrained to reproduce the local environment. The goal of this study is to inspect whether the direction of accretion of satellites on to the Milky Way and Andromeda galaxies is related to the cosmic web. The analysis considers the three high-resolution simulations available in the HESTIA simulation suite, as well as the derived velocity shear and tidal tensors. We notice two eras in the LG accretion history, delimited by an epoch around z ≍ 0.7. We also find that satellites can travel up to ~4 Mpc, relative to their parent halo before crossing its viral radius R200. Finally, we observe a strong alignment of the infall direction with the axis of slowest collapse $\boldsymbol{e}_{3}$ of both tidal and shear tensors, implying satellites of the LG originated from one particular region of the cosmic web and were channeled towards us via the process of accretion.This alignment is dominated by the satellites that enter during the early infall era, i.e. z > 0.7.