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Isadora, Hi.

I recreated your image. All vector.

I made that stroke rounded rectangle. With the white arrow, I selected to see the real handles.

I took a screenshot so I could recreate those graphic handles. I did one, dup and modified to create the others.

I drew out some Guides so I would know the perfect placement.

Lastly, I place on a background rectangle.

K

There is a script that may do what you want:

jooHandlesToGeometry

This script converts the Bézier manipulator handles to geometry. This can be useful if you need to publish tutorials, especially for print. It might also be used to customize the look and feel.

it can be found here:

https://bitbucket.org/joojaa/jooillustratorscripts/src/master/

You want to create this as an artwork?

And it should be animated? 

While tehre are scripts to create the handles as objects, this cannot be animated. Because when animated it would need to change dynamically.