The gravitational-wave event GW170817 together with the electromagnetic counterpart showed that the speed of tensor perturbations cT on the cosmological background is very close to that of light c for the
redshift z < 0.009. In generalized Proca theories, the Lagrangians compatible with the condition cT = c are constrained to be derivative interactions up to cubic order besides those corresponding to intrinsic
vector modes. We place observational constraints on a dark energy model in cubic-order generalized Proca theories with intrinsic vector modes by running the Markov-Chain- Monte-Carlo (MCMC) code. We use the cross-correlation data of the integrated Sachs-Wolfe (ISW) signal and galaxy distributions in addition to the datasets of cosmic microwave background, baryon acoustic oscillations, type Ia supernovae, local
measurements of the Hubble expansion rate, and redshift-space distortions. We show that, unlike cubic-order scalar-tensor theories, the existence of intrinsic vector modes allows the possibility for
evading the ISW-galaxy anti-correlation incompatible with the current observational data. As a result, we find that the dark energy model in cubic-order generalized Proca theories exhibits a better fit to the data
than the cosmological constant even by including the ISW-galaxy correlation data in the MCMC analysis.