This letter presents vision-based autonomous navigation of a steerable soft growing robot. Our experimental platform is the previously presented MAMMOBOT, which is a small-diameter eversion growing robot with an embedded steerable catheter. The current manuscript first models the robot using kinematics (constant curvature) and mechanics (virtual work). Modelling considers the potential misalignment between the everting sheath and the embedded catheter. Second, a switching control architecture is proposed, wherein a model-based controller is employed for rapid convergence to a target position, followed by a closed-loop proportional controller that minimises the system's steady-state error. Feedback is visually provided from a calibrated stereo vision system. Target-positioning and trajectory-tracking experiments are conducted to evaluate the performance of the control architecture. Experimental results demonstrate the superiority of the mechanics-based modelling and control approach, showing an average accuracy of 0.67mm (0.66% arclength) in target positioning experiments, and an accuracy of 0.72mm (1.11% arclength) and 0.72mm (1.01% arclength) for tracking a square trajectory and a circular trajectory, respectively. The autonomous steering framework is showcased within a 3D-printed mammary duct phantom. This work sets the stage for endoscope-based autonomous navigation of MAMMOBOT and similar soft growing steerable robots.