Using e^{+}e^{-} annihilation data corresponding to an integrated luminosity of 6.32 fb^{-1} collected at center-of-mass energies between 4.178 and 4.226 GeV with the BESIII detector, we perform the first amplitude analysis of the decay D_{s}^{+}→K_{S}^{0}K^{+}π^{0} and determine the relative branching fractions and phases for intermediate processes. We observe an a_{0}-like state with mass of 1.817 GeV in its decay to K_{S}^{0}K^{+} for the first time. In addition, we measure the ratio {B[D_{s}^{+}→K[over ¯]^{*}(892)^{0}K^{+}]/B[D_{s}^{+}→K[over ¯]^{0}K^{*}(892)^{+}]} to be 2.35_{-0.23stat}^{+0.42}±0.10_{syst}. Finally, we provide a precision measurement of the absolute branching fraction B(D_{s}^{+}→K_{S}^{0}K^{+}π^{0})=(1.46±0.06_{stat}±0.05_{syst})%.