We present the first amplitude analysis and branching fraction measurement of the decay D_{s}^{+}→π^{+}π^{+}π^{-}π^{0}π^{0}, using e^{+}e^{-} collision data collected with the BESIII detector at center-of-mass energies between 4.128 and 4.226 GeV corresponding to an integrated luminosity of 7.33 fb^{-1}, and report the first observation of the pure W-annihilation decay D_{s}^{+}→ωρ^{+} with a branching fraction of (0.99±0.08_{stat} _{-0.07}^{+0.05}_{syst})%. In comparison to the low significance of the D wave in the decay D_{s}^{+}→ϕρ^{+}, the dominance of the D wave over the S and P waves, with a fraction of (51.85±7.28_{stat} _{-7.90}^{+4.83}_{syst})% observed in the decay D_{s}^{+}→ωρ^{+}, provides crucial information for the "polarization puzzle," as well as for the understanding of charm meson decays. The branching fraction of D_{s}^{+}→π^{+}π^{+}π^{-}π^{0}π^{0} is measured to be (4.41±0.15_{stat}±0.13_{syst})%. Moreover, the branching fraction of D_{s}^{+}→ϕρ^{+} is measured to be (3.98±0.33_{stat} _{-0.19}^{+0.21}_{syst})%, and the R_{ϕ}=B(ϕ→π^{+}π^{-}π^{0})/B(ϕ→K^{+}K^{-}) is determined to be (0.222±0.019_{stat} _{-0.016}^{+0.016}_{syst}), which is consistent with the previous measurement based on charm meson decays, but deviates from the results from e^{+}e^{-} annihilation and K-N scattering experiments by more than 3σ.