Laser weeding is a non-chemical, non-contact precision weeding method particularly suited for inter-row operations during the crop seedling stage. However, maintaining precise laser targeting on weed stems during dynamic field operation remains challenging. This task requires close coordination among perception, control, and actuation modules, while vehicle vibration from uneven terrain further degrades strike precision on the move. This paper presents a fully autonomous laser weeding robot for inter-row operation in cotton seedling fields, integrating weeding control, automatic mode switching, and autonomous navigation into a coordinated multi-process architecture. Vibration-induced strike deviation is mitigated by STCF-Net (Spatio-Temporal Cross-modal Fusion Network), a cross-modal prediction framework that fuses forward-looking depth maps with historical IMU signals to forecast vibration severity over a 3 s horizon. The prediction enables the control system to proactively adjust travel speed and weeding mode before encountering rough terrain, balancing precision and efficiency under dynamic field conditions. Field experiments validated weeding performance in static and dynamic scenarios and demonstrated that STCF-Net-guided predictive switching reduced operational time by 11.5% to 13.9% compared with a conservative baseline in two controlled experiments, with only a marginal reduction in weeding precision. This study reports the system-level design of a fully autonomous laser weeding robot and introduces feedforward vibration prediction to the laser weeding domain, providing a practical reference for integrating predictive algorithms into field-deployable weeding systems.