Hi JessWThomas,
This is certainly one of the more complex questions, to do it with any kind of rig from “A to Z”. To do the step at 07:30 [clip/min:sec], with a paper like stiffness feel to it, would take a while by itseld.
Have a look at the set-up for a simple folding of a package wrap:
https://www.cineversity.com/forums/viewthread/2206/
Since there were only a limited folds I suggested to think of all steps and include them. With Origami, as in your example, I doubt that would be useful.
My best guess for a complete animation of this (like in your clip-example) would be in a step by step. After each step one would take the current state, delete the rig, and place new joints and weighting to it. of course each time with a copy of the previous. That perhaps a dozen of times.
The initial step would be, from where I look at it, to fold the practical paper until it is pretty clear (muscle-memory) how it works.
The created folds need to be in the mesh. What might be even more difficult would be the pulling of folds to gain volume, as seen in other examples of this art form.
I don’t think it is impossible, but it is certainly not something that is quickly done, and it needs for each a new approach.
My step by step suggestion allows to limit the complexity per step. Once a step is done, Current State to Object and create a rig again.
Since we can copy models/objects and start over without visible change, this would be one way.
Perhaps simpler and my way of doing would be do edit it in steps and move the camera while changing from “step to step”.
This certainly needs a good film-editing.
Joints, Pose Morph, and other options might be useable, Cluster, Correction and other Deformer of course.
The practical paper used for this could be a nice texture, and each step as well scanned and animated as a sequence to allow for a more convincing result perhaps.
My best wishes
