  We study a vector dark matter (VDM) model in which the dark sector couples to
the Standard Model sector via a Higgs portal. If the portal coupling is small
enough the VDM can be produced via the freeze-in mechanism. It turns out that
the electroweak phase transition have a substantial impact on the prediction of
the VDM relic density. We further assume that the dark Higgs boson which gives
the VDM mass is so light that it can induce strong VDM self-interactions and
solve the small-scale structure problems of the Universe. As illustrated by the
latest LUX data, the extreme smallness of the Higgs portal coupling required by
the freeze-in mechanism implies that the dark matter direct detection bounds
are easily satisfied. However, the model is well constrained by the indirect
detections of VDM from BBN, CMB, AMS-02, and diffuse $\gamma$/X-rays.
Consequently, only when the dark Higgs boson mass is at most of ${\cal O}({\rm
keV})$ does there exist a parameter region which leads to a right amount of VDM
relic abundance and an appropriate VDM self-scattering while satisfying all
other constraints simultaneously.
