Robotic autonomy at centimeter scale requires compact and miniaturization-friendly actuation integrated with sensing and neural network processing assembly within a tiny form factor. Applications of such systems have witnessed significant advancements in recent years in fields such as healthcare, manufacturing, and post-disaster rescue. The system design at this scale puts stringent constraints on power consumption for both the sensory front-end and actuation back-end and the weight of the electronic assembly for robust operation. In this paper, we introduce FAVbot, the first autonomous mobile micro-robotic system integrated with a novel actuation mechanism and convolutional neural network (CNN) based computer vision - all integrated within a compact 3-cm form factor. The novel actuation mechanism utilizes mechanical resonance phenomenon to achieve frequency-controlled steering with a single piezoelectric actuator. Experimental results demonstrate the effectiveness of FAVbot's frequency-controlled actuation, which offers a diverse selection of resonance modes with different motion characteristics. The actuation system is complemented with the vision front-end where a camera along with a microcontroller supports object detection for closed-loop control and autonomous target tracking. This enables adaptive navigation in dynamic environments. This work contributes to the evolving landscape of neural network-enabled micro-robotic systems showing the smallest autonomous robot built using controllable multi-directional single-actuator mechanism.