As in the case of the Smith predictor, when applied to dynamical systems with delay in the input, the stability of the internal model control (IMC) system can be very sensitive to the mismatch that can occur between the plant delay and the identified delay. In this paper, we propose an online identification technique for the plant with input delay, which is used to adapt an IMC controller designed to reject periodic disturbances consisting of two harmonics. The proposed method is based on wavelet analysis and cross-correlation of the measured output. Moreover, the proposed controller structure introduces a novel analytical approach which makes it suitable for the adaptive application. The practical motivation of the study is to address the delay mismatch that can be caused by approximating time-varying high-order systems by first-order models with input delay. The approach is numerically validated.