Mobile cable-driven parallel robots (MCDPRs) offer expanded motion capabilities and workspace compared to traditional cable-driven parallel robots (CDPRs) by incorporating mobile bases. However, additional mobile bases introduce more degree-of-freedom (DoF) and various constraints to make their motion planning a challenging problem. Despite several motion planning methods for MCDPRs being developed in the literature, they are only applicable to known environments, and autonomous navigation in unknown environments with obstacles remains a challenging issue. The ability to navigate autonomously is essential for MCDPRs, as it opens up possibilities for the robot to perform a broad range of tasks in real-world scenarios. To address this limitation, this study proposes an online motion planning method for MCDPRs based on the pipeline of rapidly exploring random tree (RRT). The presented approach explores unknown environments efficiently to produce high-quality collision-free trajectories for MCDPRs. To ensure the optimal execution of the planned trajectories, the study introduces two indicators specifically designed for the mobile bases and the end-effector. These indicators take into account various performance metrics, including trajectory quality and kinematic performance, enabling the determination of the final following trajectory that best aligns with the desired objectives of the robot. Moreover, to effectively handle unknown environments, a vision-based system utilizing an RGB-D camera is developed, allowing for precise MCDPR localization and obstacle detection, ultimately enhancing the autonomy and adaptability of the MCDPR. Finally, the extensive simulations conducted using dynamic simulation software (CoppeliaSim) and the on-board real-world experiments with a self-built MCDPR prototype demonstrate the practical applicability and effectiveness of the proposed method.