  Using a data sample of $(1310.6 \pm 7.0)\times 10^6$ $J/\psi$ events
collected with the BESIII detector operating at the BEPCII collider, we perform
the first experimental search for invisible decays of a light vector meson
($V=\omega,\phi$) via $J/\psi \to V \eta$ decays. The decay of $\eta \to
\pi^+\pi^-\pi^0$ is utilized to tag the $V$ meson decaying into the invisible
final state. No evidence for a significant invisible signal is observed, and
the upper limits on the ratio of branching fractions at the 90\% confidence
level are determined to be $\frac{\mathcal{B}(\omega \rightarrow
\rm{invisible})}{\mathcal{B}(\omega \to \pi^+\pi^-\pi^0)} < 8.1 \times 10^{-5}$
and $\frac{\mathcal{B}(\phi \to \rm{invisible})}{\mathcal{B}(\phi \to K^+K^-)}
< 3.4\times 10^{-4}$. By using the world average values of $\mathcal{B}(\omega
\to \pi^+\pi^-\pi^0)$ and $\mathcal{B}(\phi\to K^+K^-)$, the upper limits on
the decay branching fractions at the $90\%$ confidence level are set as
  $\mathcal{B}(\omega \to \rm{invisible})< 7.3 \times 10^{-5}$ and
$\mathcal{B}(\phi \to \rm{invisible})< 1.7\times 10^{-4}$, respectively.
