I cannot use live action animation, because I'm too disabled.

Yes, there are a few ways to animate without recording your performance with a webcam.

You could add a transform behavior to the layers (or handles?) and using keyframes to animate them. You then set them up as replays that you can trigger. 

You could use the dragger behavior to move things around. 

If you just want some really basic movement, you could try adding a handle to the head and applying the Dangle tag to it, playing with the settings until it doesn't sink or squash too much (maybe set the gravity / force to go up instead of down) and adding wind to make it sway. There's an auto/blink behavior for making eyes blink automatically.

You can also animate frames for just about any layer and put those frames in as layers and set them to cycle layers. You could then trigger those layers to play back when you want. You might also be able to use the layer picker behavior to control which frame of the animation to play.

If you have a webcam and someone else who's willing to help, you could have them act out different movements/expressions and then make some replays from their performance and make them triggerable as well. 

 I think you can do it, but not in the way puppets normally work. Basically you don't use any behavior that relies on the webcam. Just turn the webcam off.

For example, you can use the mouse to control eyes. You can put Draggers and transforms on eyebrows to move them. You can use cycle layers for other expressions from a trigger. You can create replays so you set up certain expressions once with key frames or similar (using mouse) then trigger them. With replays in place I think you may even be able to do a live recording from a fixed set of movements.

The webcam I find a mixed blessing. For good recordings I turn off eye snap and web cam and use a mouse. Even eyebrows I tend to not use webcam as they jitter a bit (maybe it's just my camera and lighting). It is really just head tilt that I use the webcam for, and id puppet set up carefully you could use a transform behavior and rotation to do a reasonable job.

live performances will be harder, but if prerecorded is fine then i think you can do it - as long as you have reasonable mouse skills. Oh, and good eyesight at times! sometimes i have to squint to see some of the small controls you need to click on for like key frames.

Yup, that makes perfect sense, but unfortunately this is not currently a feature. It would be pretty awesome, though, so it's worth posting a feature request on the Character Animator UserVoice page: https://adobe-video.uservoice.com/forums/911317-character-animator