In this paper, we propose and experimentally demonstrate a distance-based rate-adaptive visible light communication (VLC) system based on constellation probabilistic shaping (PS) for a multiple-user access network. For users with different access distance, we optimize the transmission data rate close to the channel capacity by applying PS combined with code-rate adaptive FEC at the transmitter side according to the per-user signal-to-noise ratio (SNR) budget. This is also proved to be a convenient way to ensure fine granularity of information rate per user with wider flexibility compared with non-PS modulation formats. We also investigate the performances of different PS-QAM modulation formats under different SNR level when considering peak-to-average power ratio (PAPR) in the VLC system. Optimal PS-QAM and FEC code-rate are also studied in the flexible VLC access system. In addition, in order to overcome the nonlinear distortion in the system, a neural network (NN) is used as the post-equalization. Finally, we demonstrate the flexible access with the net data-rate from 1.84 to 3.37 Gbps for 20 and 1-meter distance, with a maximum 28% overall capacity improvement compared with regular non-PS modulations.