We present the results of a series of cosmological N-body simulations of a vector dark energy (VDE) model, performed using a suitably modified version of the publicly available gadget-2 code. The set-ups of our simulations were calibrated pursuing a twofold aim: (1) to analyse the large-scale distribution of massive objects and (2) to determine the properties of halo structure in this different framework. We observe that structure formation is enhanced in VDE, since the mass function at high redshift is boosted up to a factor of 10 with respect to Λ cold dark matter (ΛCDM), possibly alleviating tensions with the observations of massive clusters at high redshifts and early reionization epoch. Significant differences can also be found for the value of the growth factor, which in VDE shows a completely different behaviour, and in the distribution of voids, which in this cosmology are on average smaller and less abundant. We further studied the structure of dark matter haloes more massive than 5 × 1013 h−1 M⊙, finding that no substantial difference emerges when comparing spin parameter, shape, triaxiality and profiles of structures evolved under different cosmological pictures. Nevertheless, minor differences can be found in the concentration–mass relation and the two-point correlation function, both showing different amplitudes and steeper slopes. Using an additional series of simulations of a ΛCDM scenario with the same and σ8 used in the VDE cosmology, we have been able to establish whether the modifications induced in the new cosmological picture were due to the particular nature of the dynamical dark energy or a straightforward consequence of the cosmological parameters. On large scales, the dynamical effects of the cosmic vector field can be seen in the peculiar evolution of the cluster number density function with redshift, in the shape of the mass function, in the distribution of voids and on the characteristic form of the growth index γ(z). On smaller scales, internal properties of haloes are almost unaffected by the change of cosmology, since no statistical difference can be observed in the characteristics of halo profiles, spin parameters, shapes and triaxialities. Only halo masses and concentrations show a substantial increase, which can, however, be attributed to the change in the cosmological parameters.