A symmetry-preserving approach to the two valence-body continuum bound-state problem is used to calculate the elastic electromagnetic form factors of the rho-meson and subsequently to study the evolution of vector-meson form factors with current-quark mass. To facilitate a range of additional comparisons, K* form factors are also computed. The analysis reveals that: vector mesons are larger than pseudoscalar mesons; composite vector mesons are non-spherical, with magnetic and quadrupole moments that deviate ~30% from point-particle values; in many ways, vector-meson properties are as much influenced by emergent mass as those of pseudoscalars; and vector meson electric form factors possess a zero at spacelike momentum transfer. Qualitative similarities between the electric form factors of the rho and the proton, GpE, are used to argue that the character of emergent mass in the Standard Model can force a zero in GpE. Moreover, the existence of a zero in vector meson electric form factors entails that a single-pole vector meson dominance model can only be of limited use in estimating properties of off-shell vector mesons, providing poor guidance for systems in which the Higgs-mechanism of mass generation is dominant.