This article considers a class of noncooperative games in which the feasible decision sets of all players are coupled together by a coupled inequality constraint. Adopting the variational inequality formulation of the game, we first introduce a new local edge-based equilibrium condition with full information. Considering challenges when communication delays occur, we then devise an asynchronous distributed algorithm to seek a generalized Nash equilibrium. This asynchronous scheme arbitrarily activates one player to start new computations independently at different iteration instants, which means that the picked player can use the involved outdated information from itself and its neighbors to perform new updates. In theoretical aspect, we provide explicit conditions on algorithm parameters, for instance, the step-sizes to establish a sublinear convergence rate for the synchronous version. Next, the asynchronous algorithm guarantees almost sure convergence in expectation under the same step-size conditions and some standard assumptions. Finally, the viability and performance of the proposed algorithm are demonstrated by numerical studies on the Cournot competition.