In frequency-division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems, downlink channel state information (CSI) needs to be sent from users back to the base station (BS), which causes prohibitive feedback overhead. In this paper, we propose a lightweight and adaptive deep learning-based CSI feedback scheme by capitalizing on deep equilibrium models. Different from existing deep learning-based approaches that stack multiple explicit layers, we propose an implicit equilibrium block to mimic the process of an infinite-depth neural network. In particular, the implicit equilibrium block is defined by a fixed-point iteration and the trainable parameters in each iteration are shared, which results in a lightweight model. Furthermore, the number of forward iterations can be adjusted according to the users' computational capability, achieving an online accuracy-efficiency trade-off. Simulation results will show that the proposed method obtains a comparable performance as the existing benchmarks but with much-reduced complexity and permits an accuracy-efficiency trade-off at runtime.