Succinonitrile (N C-CH2-CH2-C N), an orientationally disordered molecular plastic crystal at room temperature, exhibits rich phase behavior including a solid-solid phase transition at 238 K. In cooling through this phase transition, the high-temperature rotational disorder of the plastic crystal phase is frozen out, forming a rigid crystal that is both spatially and orientationally ordered. Using temperature-dependent terahertz time-domain spectroscopy, we characterize the vibrational modes of this low-temperature crystalline phase for frequencies from 0.3 to 2.7 THz and temperatures ranging from 20 to 220 K. Vibrational modes are observed at 1.122 and 2.33 THz at 90 K. These modes are assigned by solid-state density functional theory simulations, corresponding respectively to the translation and rotation of the molecules along and about their crystallographic c-axis. In addition, we observe a suppression of the phonon modes as the concentration of dopants, in this case a lithium salt (LiTFSI), increases, indicating the importance of doping-induced disorder in these ionic conductors.