Ammonium scheelites (of which ammonium perrhenate is an example) show several anomalies in their behaviour (including thermal contraction of the a lattice parameter, large thermal expansion of the c lattice parameter, enhanced nuclear quadrupolar relaxation, increased specific heat) which has been attributed to the onset of orientational disorder of the ammonium ions. Unlike in the ammonium halides there is no actual phase transition. Extensive molecular dynamics simulations of this material are reported. The model shows anomalies in the thermal expansion which are associated with a broadening and shift of the orientational distribution of the ammonium ions. There is no evidence of more than one preferred orientation. The ammonium ions also begin to jump between the twelve equivalent orientations, and we present evidence that this is primarily by rotation about one of the bonds. The activation energy for this agrees well with experiment. Angular and linear velocity-time correlation functions for the ammonium ion show rapid dephasing at higher temperatures, which is consistent with the observed broadening of the corresponding lines in the Raman spectrum. The origin of the anomalous thermal expansion is discussed in terms of various contributions to the entropy of the crystal.