Cosmological N-body simulations show that Milky-Way-sized galaxies harbor a population of unmerged dark matter subhalos. These subhalos could shine in gamma rays and be eventually detected in gamma-ray surveys as unidentified sources. We search for very-high-energy (VHE, $E\geq 100$ GeV) gamma-ray emission using H.E.S.S. observations carried out from a thorough selection of unidentified Fermi-LAT Objects (UFOs) as dark matter subhalo candidates. Provided that the dark matter mass is higher than a few hundred GeV, the emission of the UFOs can be well described by dark matter annihilation models. No significant VHE gamma-ray emission is detected in any UFO dataset nor in their combination. We, therefore, derive constraints on the product of the velocity-weighted annihilation cross-section $\langle \sigma v\rangle$ by the $J$-factor on dark matter models describing the UFO emissions. Upper limits at 95% confidence level are derived on $\langle \sigma v\rangle J$ in $W^+W^-$ and $\tau^+\tau^-$ annihilation channels for the TeV dark matter particles. Focusing on thermal WIMPs, strong constraints on the $J$-factors are obtained from H.E.S.S. observations. Adopting model-dependent predictions from cosmological N-body simulations on the $J$-factor distribution function for Milky Way (MW)-sized galaxies, only $\lesssim 0.3$ TeV mass dark matter models marginally allow to explain observed UFO emission.