It is entirely possible. Of course your character shapes will be limited and the sturdiness of your puppet will depend only on how hard or soft the clay is.
I haven't tried using armatures in any of my videos yet, though I am in the process of making a couple of wire armatures to try out.
I usually have to bulk up the lower part of the puppet for stabilization. On a bipedal puppet, this generally would mean making thick and heavy legs and feet.
I find this makes the puppet more bottom heavy and less likely to sag or tip over.
Also the clay cant be too soft or you will have floppy problems. Hope that helps.