This paper introduces a reconfigurable intelligent surface (RIS)-assisted grant-free non-orthogonal multiple-access (GF-NOMA) scheme. To ensure the power reception disparity required by the power domain NOMA (PD-NOMA), we propose a joint user clustering and RIS assignment/alignment approach that maximizes the network sum rate by judiciously pairing user equipments (UEs) with distinct channel gains, assigning RISs to proper clusters, and aligning RIS phase shifts to the cluster members yielding the highest cluster sum rate. Once UEs are acknowledged with the cluster index, they are allowed to access their resource blocks (RBs) at any time requiring neither further grant acquisitions from the base station (BS) nor power control as all UEs are requested to transmit at the same power. In this way, the proposed approach performs an implicit over-the-air power control with minimal control signaling between BS and UEs, which has shown to deliver up to 20% higher network sum rate than benchmark GF-NOMA and optimal grant-based PD-NOMA schemes depending on the network parameters. The given numerical results also investigate the impact of UE density, RIS deployment, and RIS hardware specifications on the overall performance of the proposed RIS-aided GF-NOMA scheme.