Orbital angular momentum (OAM) technology has attracted much research interest in recent years because of its characteristic helical phase front twisting around the propagation axis and natural orthogonality among different OAM states to encode more degrees of freedom than classical planar beams. Leveraging upon these features, OAM technique has been applied to wireless communication systems to enhance spectral efficiency and radar systems to distinguish spatial targets without beam scanning. Leveraging upon these unique properties, we propose an OAM-based millimeter-wave joint radar-communications (JRC) system comprising a bi-static automotive radar and vehicle-to-vehicle (V2V) communications. Different from existing uniform circular array (UCA) based OAM systems where each element is an isotropic antenna, an OAM spatial modulation scheme utilizing a uniform linear array (ULA) is adopted with each element being a traveling-wave antenna, producing multiple Laguerre-Gaussian (LG) vortex beams simultaneously. Specifically, we first build a novel bi-static automotive OAM-JRC model that embeds communication messages in a radar signal, following which a target position and velocity parameters estimation algorithm is designed with only radar frames. Then, an OAM-based mode-division multiplexing (MDM) strategy between radar and JRC frames is presented to ensure the JRC parameters identifiability and recovery. Furthermore, we analyze the performance of the JRC system through deriving recovery guarantees and Cramér-Rao lower bound (CRLB) of radar target parameters and evaluating the bit error rate (BER) of communication, respectively. Our numerical experiments validate the effectiveness of the proposed OAM-based JRC system and parameter estimation method.