Is this possible in Blender?

Is it possible to do this in Blender? If so, do you know of any instructions or tutorials online that would show me how to do it?

I want to be able to move a character around but then have physics animate the secondary action of the surface of the object.

You have an addon called Wiggle Bones:

You could also use either cloth or soft bodies, just create a vertex group to define the parts of the model you want to have the jiggle effect and bake the simulation. With this method you can have a very good looking secondary motion. Maybe not realtime (depending on your machine) but pretty good anyway

Here’s a video showing the process:

Thanks for the response @julperado !
That will be very helpful in some situations I’m currently working on. I didn’t realize I could weight paint the vertex group for cloth.

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