In this paper, we identify optical reconfigurable intelligent surfaces (ORISs) as key enablers of next-generation free-space optical (FSO) multiple access systems. By leveraging their beam steering and beam splitting capabilities, ORISs are able to effectively address line-of-sight (LoS) constraints, while enabling multi-user connectivity. We consider an ORIS-assisted non-orthogonal multiple access (NOMA) system model consisting of a single transmitter (Tx) and two receivers (Rxs). We derive novel analytical expressions to characterize the statistical particularities of the Tx-ORIS-Rx communication channel. Building upon the aforementioned expressions, we investigate the outage performance of the Rxs by deriving exact analytical expressions for the outage probability (OP) of each Rx. To provide deeper insights into the impact of various system parameters and physical conditions on the outage performance of each Rx, we conduct a high signal-to-noise ratio (SNR) analysis, that returns asymptotic expressions for the Rxs OPs at the high-SNR regime. Monte Carlo simulations validate the analysis, demonstrate the effectiveness of ORIS-enabled NOMA under a variety of configurations and physical scenarios, and showcase its superiority over its orthogonal-based counterpart.