The gravitational-wave event GW170817, together with the electromagnetic counterpart, shows that the speed of tensor perturbations $c_T$ 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 $c_T=c$ 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 data sets 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 anticorrelation 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.
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