Creating animation takes time, effort, and technical expertise. To help novices with animation, we present LogoMotion, an AI code generation approach that helps users create semantically meaningful animation for logos. LogoMotion automatically generates animation code with a method called visually-grounded code synthesis and program repair. This method performs visual analysis, instantiates a design concept, and conducts visual checking to generate animation code. LogoMotion provides novices with code-connected AI editing widgets that help them edit the motion, grouping, and timing of their animation. In a comparison study on 276 animations, LogoMotion was found to produce more content-aware animation than an industry-leading tool. In a user evaluation (n=16) comparing against a prompt-only baseline, these code-connected widgets helped users edit animations with control, iteration, and creative expression.