Recently, reconfigurable intelligent surface (RIS) has gained research and development interests to modify wireless channel characteristics in order to improve performance of wireless communications, especially when quality of the line-of-sight channel is not that good. In this work, for the first time in the literature, we have used simultaneously transmitting and reflecting RIS (STAR-RIS) in non-orthogonal multiple-access visible light communication system to improve performance of the system. Achievable rates of the users are derived for two data recovery schemes, single-user detection (SUD) and successive interference cancellation (SIC). Then, sum-rate optimization problem is formulated for two operating modes of STAR-RIS, namely energy-splitting and mode-switching cases. Moreover, a sequential parametric convex approximation method is used to solve the sum-rate optimization problems. We have also compared energy-splitting and mode-switching cases and showed that these two modes have the same performance. Finally, numerical results for SUD and SIC schemes and two benchmarking schemes, time-sharing and max-min fairness, are presented and spectral- and energy-efficiency, number of STAR-RIS elements, position of users and access point are discussed.