Cross sections of the process e+e-→π0π0J/ψ at center-of-mass energies between 3.808 and 4.600 GeV are measured with high precision by using 12.4 fb-1 of data samples collected with the BESIII detector operating at the BEPCII collider facility. A fit to the measured energy-dependent cross sections confirms the existence of the charmoniumlike state Y(4220). The mass and width of the Y(4220) are determined to be (4220.4±2.4±2.3) MeV/c2 and (46.2±4.7±2.1) MeV, respectively, where the first uncertainties are statistical and the second systematic. The mass and width are consistent with those measured in the process e+e-→π+π-J/ψ. The neutral charmonium-like state Zc(3900)0 is observed prominently in the π0J/ψ invariant-mass spectrum, and, for the first time, an amplitude analysis is performed to study its properties. The spin-parity of Zc(3900)0 is determined to be JP=1+, and the pole position is (3893.1±2.2±3.0)-i(22.2±2.6±7.0) MeV/c2, which is consistent with previous studies of electrically charged Zc(3900)±. In addition, cross sections of e+e-→π0Zc(3900)0→π0π0J/ψ are extracted, and the corresponding line shape is found to agree with that of the Y(4220).