This paper explores a novel integrated localization and communication (ILAC) system using the affine Fourier transform multicarrier (AFT-MC) waveform. Specifically, we consider a multiple-input multiple-output (MIMO) AFT-MC system with ILAC and derive a continuous delay and Doppler shift channel matrix model. Based on the derived signal model, we develop a two-step algorithm with low complexity for estimating channel parameters. Furthermore, we derive the Cramér-Rao lower bound (CRLB) of location estimation as the fundamental limit of localization. Finally, we provide some insights about the AFT-MC parameters by explaining the impact of the parameters on localization performance. Simulation results demonstrate that the AFT-MC waveform is able to provide significant localization performance improvement compared to orthogonal frequency division multiplexing (OFDM) while achieving the CRLB of location estimation.