An amplitude analysis of the π0π0 system produced in radiative J/ψ decays is presented. In particular, a piecewise function that describes the dynamics of the π0π0 system is determined as a function of Mπ0π0 from an analysis of the (1.311±0.011)×109 J/ψ decays collected by the BESIII detector. The goal of this analysis is to provide a description of the scalar and tensor components of the π0π0 system while making minimal assumptions about the properties or number of poles in the amplitude. Such a model-independent description allows one to integrate these results with other related results from complementary reactions in the development of phenomenological models, which can then be used to directly fit experimental data to obtain parameters of interest. The branching fraction of J/ψ→γπ0π0 is determined to be (1.15±0.05)×10-3, where the uncertainty is systematic only and the statistical uncertainty is negligible.