Visible light communication (VLC) is among the key enabling technologies to shape future wireless networks due to the unlicensed and vast amount of bandwidth in the visible light spectrum. Unlike radio frequency (RF) systems, signal transmission in VLC is severely affected by line-of-sight (LoS) link blockages, random device orientation, and limited field-of-view (FoV) of VLC transceivers, which lead to non-uniform coverage, reduced data rate and limited mobility support. Recently, the idea of movable access points (MAPs) in VLC systems has been proposed to overcome these challenges. This paper investigates the concepts of single- and multi-connectivity in MAP-based VLC systems. Specifically, two sum rate optimization frameworks for single- and multi-connectivity-enabled MAP-based VLC system are proposed, while considering minimum quality-of-service requirements for communication and sensing users and power budget. Then, the majorization-minimization approach is leveraged to develop an algorithm for the frameworks. Finally, simulation results are used to evaluate the sum rate and computational complexity of the two frameworks. The results reveal that single- and multi-connectivity frameworks mitigate the impact of LoS blockages, user mobility, and limited FoV of transceivers in VLC systems and a trade-off between complexity and sum rate, where multi-connectivity offers a 300–400% improvement, but a 2.4x increase in computational complexity compared to single-connectivity.