Using e^{+}e^{-} collision data corresponding to an integrated luminosity of 7.33 fb^{-1} recorded by the BESIII detector at center-of-mass energies between 4.128 and 4.226 GeV, we present an analysis of the decay D_{s}^{+}→π^{+}π^{-}e^{+}ν_{e}, where the D_{s}^{+} is produced via the process e^{+}e^{-}→D_{s}^{*±}D_{s}^{∓}. We observe the f_{0}(980) in the π^{+}π^{-} system and the branching fraction of the decay D_{s}^{+}→f_{0}(980)e^{+}ν_{e} with f_{0}(980)→π^{+}π^{-} measured to be (1.72±0.13_{stat}±0.10_{syst})×10^{-3}, where the uncertainties are statistical and systematic, respectively. The dynamics of the D_{s}^{+}→f_{0}(980)e^{+}ν_{e} decay are studied with the simple pole parametrization of the hadronic form factor and the Flatté formula describing the f_{0}(980) in the differential decay rate, and the product of the form factor f_{+}^{f_{0}}(0) and the c→s Cabibbo-Kobayashi-Maskawa matrix element |V_{cs}| is determined for the first time to be f_{+}^{f_{0}}(0)|V_{cs}|=0.504±0.017_{stat}±0.035_{syst}. Furthermore, the decay D_{s}^{+}→f_{0}(500)e^{+}ν_{e} is searched for the first time but no signal is found. The upper limit on the branching fraction of D_{s}^{+}→f_{0}(500)e^{+}ν_{e}, f_{0}(500)→π^{+}π^{-} decay is set to be 3.3×10^{-4} at 90% confidence level.