We report the triton (t) production in midrapidity (|y|<0.5) Au+Au collisions at sqrts_{NN}=7.7-200  GeV measured by the STAR experiment from the first phase of the beam energy scan at the Relativistic Heavy Ion Collider. The nuclear compound yield ratio (N_{t}×N_{p}/N_{d}^{2}), which is predicted to be sensitive to the fluctuation of local neutron density, is observed to decrease monotonically with increasing charged-particle multiplicity (dN_{ch}/dη) and follows a scaling behavior. The dN_{ch}/dη dependence of the yield ratio is compared to calculations from coalescence and thermal models. Enhancements in the yield ratios relative to the coalescence baseline are observed in the 0%-10% most central collisions at 19.6 and 27 GeV, with a significance of 2.3σ and 3.4σ, respectively, giving a combined significance of 4.1σ. The enhancements are not observed in peripheral collisions or model calculations without critical fluctuation, and decreases with a smaller p_{T} acceptance. The physics implications of these results on the QCD phase structure and the production mechanism of light nuclei in heavy-ion collisions are discussed.