We consider a micromanipulation problem in eye surgery, specifically retinal vein cannulation (RVC). RVC involves inserting a microneedle into a retinal vein for the purpose of targeted drug delivery. The procedure requires accurately guiding a needle to a target vein and inserting it while avoiding damage to the surrounding tissues. RVC can be considered similar to the reach or push task studied in robotics manipulation, but with additional constraints related to precision and safety while interacting with soft tissues. Prior works have mainly focused developing robotic hardware and sensors to enhance the surgeons' accuracy, leaving the automation of RVC largely unexplored. In this paper, we present the first autonomous strategy for RVC while relying on a minimal setup: a robotic arm, a needle, and monocular images. Our system exclusively relies on monocular vision to achieve precise navigation, gentle placement on the target vein, and safe insertion without causing tissue damage. Throughout the procedure, we employ machine learning for perception and to identify key surgical events such as needle-vein contact and vein punctures. Detecting these events guides our task and motion planning framework, which generates safe trajectories using model predictive control to complete the procedure. We validate our system through 24 successful autonomous trials on 4 cadaveric pig eyes. We show that our system can navigate to target veins within 22 micrometers of XY accuracy and under 35 seconds, and consistently puncture the target vein without causing tissue damage. Preliminary comparison to a human demonstrates the superior accuracy and reliability of our system.