Billions of vascular access procedures are performed annually worldwide, serving as a crucial first step in various clinical diagnostic and therapeutic procedures. For pediatric or elderly individuals, whose vessels are small in size (typically 2 to 3 mm in diameter for adults and less than 1 mm in children), vascular access can be highly challenging. This study presents an image-guided robotic system aimed at enhancing the accuracy of difficult vascular access procedures. The system integrates a 6-DoF robotic arm with a 3-DoF end-effector, ensuring precise navigation and needle insertion. Multi-modal imaging and sensing technologies have been utilized to endow the medical robot with precision and safety, while ultrasound imaging guidance is specifically evaluated in this study. To evaluate in vivo vascular access in submillimeter vessels, we conducted ultrasound-guided robotic blood drawing on the tail veins (with a diameter of 0.7 plus or minus 0.2 mm) of 40 rats. The results demonstrate that the system achieved a first-attempt success rate of 95 percent. The high first-attempt success rate in intravenous vascular access, even with small blood vessels, demonstrates the system's effectiveness in performing these procedures. This capability reduces the risk of failed attempts, minimizes patient discomfort, and enhances clinical efficiency.