Using 7.33 fb^{-1} of e^{+}e^{-} collision data collected by the BESIII detector at center-of-mass energies in the range of sqrt[s]=4.128-4.226 GeV, we search for the rare decays D_{s}^{+}→h^{+}(h^{0})e^{+}e^{-}, where h represents a kaon or pion. By requiring the e^{+}e^{-} invariant mass to be consistent with a ϕ(1020), 0.98<M(e^{+}e^{-})<1.04 GeV/c^{2}, the decay D_{s}^{+}→π^{+}ϕ, ϕ→e^{+}e^{-} is observed with a statistical significance of 7.8σ, and evidence for the decay D_{s}^{+}→ρ^{+}ϕ, ϕ→e^{+}e^{-} is found for the first time with a statistical significance of 4.4σ. The decay branching fractions are measured to be B(D_{s}^{+}→π^{+}ϕ,ϕ→e^{+}e^{-})=(1.17_{-0.21}^{+0.23}±0.03)×10^{-5}, and B(D_{s}^{+}→ρ^{+}ϕ,ϕ→e^{+}e^{-})=(2.44_{-0.62}^{+0.67}±0.16)×10^{-5}, where the first uncertainties are statistical and the second systematic. No significant signals for the three four-body decays of D_{s}^{+}→π^{+}π^{0}e^{+}e^{-}, D_{s}^{+}→K^{+}π^{0}e^{+}e^{-}, and D_{s}^{+}→K_{S}^{0}π^{+}e^{+}e^{-} are observed. For D_{s}^{+}→π^{+}π^{0}e^{+}e^{-}, the ϕ mass region is vetoed to minimize the long-distance effects. The 90% confidence level upper limits set on the branching fractions of these decays are in the range of (7.0-8.1)×10^{-5}.