Effectively, I'm just looking for something that would allow me to have a paper doll bounce around. Any ability to add animation to the face or whatnot isn't super important. I'm hoping there's something simple and real-time out there, since it would be a lot less time consuming to just shake a virtual stick while talking than to try to tween animation to match a voice in post.
There's Adobe Character Animator. It does pretty much exactly what you're looking for. It's a commercial product, part of the Adobe Creative Cloud suite. I'm not aware of any other real-time 2D animation packages at this time.