This work reports a detailed infrared reflectivity investigation of the phase transitions in single crystals of sodium ammonium sulfate dihydrate (SASD). The polarized reflectivity spectra allow us to follow the temperature dependence of the polar vibrational modes and detect the critical behaviour of the vibrational parameters through the two low temperature structural phase transitions observed in the compound. The results obtained show that the mechanism of the transitions in SASD is complex, involving a strong coupling between pseudo-spins and phonons. In the paraelectric phase, the driving mechanism of the first phase transition (Tc1 = 95 K) appears to be related to a relaxation with a characteristic frequency that is much lower than the phonon frequencies. In the temperature range corresponding to the first ferroelectric phase (Tc1 > T > Tc2 = 79 K), the dynamics of the lattice change considerably and the parameters characterizing several vibrational modes display anomalous temperature dependences. The second phase transition occurring at Tc2 is marked by an important and discontinuous change of the spectral shape, indicating that a considerable lattice distortion is involved.