In this paper, an active intelligent omni-surface (A-IOS) is deployed to aid uplink transmissions in a non-orthogonal multiple access (NOMA) system. In order to shelter the covert signal embedded in the superposition transmissions, a multi-antenna full-duplex (FD) receiver is utilized at the base-station to recover signal in addition to jamming the warden. With the aim of maximizing the covert rate, the FD transmit and receive beamforming, A-IOS refraction and reflection beamforming, NOMA transmit power, and FD jamming power are jointly optimized. To tackle the non-convex covert rate maximization problem subject to the highly coupled system parameters, an alternating optimization algorithm is designed to iteratively solve the decoupled sub-problems of optimizing the system parameters. The optimal solutions for the sub-problems of the NOMA transmit power and FD jamming power optimizations are derived in closed-form. To tackle the rank-one constrained non-convex fractional programming of the A-IOS beamforming and FD beamforming, a penalized Dinkelbach transformation approach is proposed to resort to the optimal solutions via semidefinite programming. Numerical results clarify that the deployment of the A-IOS significantly improves the covert rate compared with the passive-IOS aided uplink NOMA system. It is also found that the proposed scheme provides better covert communication performance with the optimized NOMA transmit power and FD jamming power compared with the benchmark schemes.