Does AE have a similar widget like the corner widget in Illustrator to easily animate corners to a rounded shape?
No, After Effects has no such widget.
Shape layers do have some control over corners, but you need to drill down in to the layer's properties to change them...
Rectangles have a "roundness" property on the path which controls the corner radius. Polygons have an "outer roundness" property which doesn't control the corner radius but the roundness of the straight segments (more like a pucker/bloat in Illustrator). Other shapes (e.g. anything manually drawn with the pen tool) have no control at all.
Shape creation in After Effects is nowhere near as complete as in Illustrator... It's often easier to just create your artwork in Illustrator and import that to After Effects (you're still free to update your AI files while working on them in After Effects).
There may be plugins for After Effects that will give you the functionality you're looking for (there are similar plugins for adding AI-like stroke control for example), so that may be worth investigating.