Single-stage single-phase onboard electric vehicle chargers have been widely adopted because of their compact structure and high efficiency. However, the performance of grid-connected operation is highly sensitive to variations in grid impedance, and this may severely compromise stability and current regulation performance. In practice, the distribution networks have uncertain and time-varying grid impedance, which, along with the charger's L-filter impedance, presents significant challenges to conventional controller designs based on nominal system parameters. This paper presents a robust inner-loop current control strategy for the single-stage single-phase onboard EV charger that explicitly takes into account both L-filter dynamics and uncertain grid impedance. The uncertain grid impedance is incorporated into the plant's representation by a small-signal model of the charger, and then, based on this model, an LMI-based robust control framework is employed to synthesize the inner-loop controller in such a way that closed-loop stability and robust performance are guaranteed over a prescribed range of grid conditions, without online estimation of grid impedance or any adaptive mechanisms. To enable implementation in the synchronous rotating reference frame, an all-pass filter is adopted to generate an artificial orthogonal (β) signal from single-phase grid voltage and current signals, and thus to facilitate dq-frame transformation and control. The proposed approach now meets a systematic design methodology for reducing the sensitivity to parameter variations while maintaining implementation simplicity. Additionally, the robust control synthesis is performed by enforcing linear matrix inequality constraints at all vertices of the polytopic uncertainty model. Specifically, sixteen vertex systems, corresponding to different combinations of L-filter parameter variations and grid impedance uncertainty, are incorporated into the LMI-based optimization, where the optimization objective is to minimize the convergence time while guaranteeing closed-loop stability and robust performance over the entire uncertainty domain. Simulation results under various grid impedance conditions confirm stable operation and reliable current tracking performance, demonstrating the effectiveness of the proposed dq-frame LMI-based robust control strategy for single-stage single-phase onboard EV chargers operating under uncertain grid conditions.