Cr(5+)-based peroxychromates, M3Cr(O2)4, with M = NH4 or a mixed NH4-alkali metal are a new class of multiferroics for potential use in molecular memory devices, with the NH4+ being a key element, but the underlying chemical mechanism is not fully understood. The NH4+ ion occupies two different sites, but their specific roles are not known. We thus performed detailed 1H NMR spin-relaxation (T1) measurements on (NH4)3Cr(O2)4 over a wide temperature range (120-300 K) to probe the displacive as well as hindered rotational dynamics of the NH4+ ions with the view of understanding their specific roles in the phase transitions. The NH4+ dynamics is seen to consist of at least three different processes with varying activation energies. The sharp jump in the T1 at around 250 K is assigned to the change in the displacive motion at one of the two sites, while a kink around 140 K is ascribed to motional slowing at the second site. Interestingly, the slowing down starts around 250 K, well above the structural phase transition at 140 K. Taken together, these results provide a clue to the role of the site and symmetry of the NH4+ ion in the mechanism of solid-solid phase transitions.