What you need to do is make a base to practice with and build your puppet on.
(Later you can fit things to it)
I hope I can explain this simply with just words.
The simplest method is cut two semi circles of stiff card/ plastic packaging. ( 2 x D shapes)
Lay them flat and put the straight edges against each other, so they form a sort of egg shape. Now separate them slightly.
Make a cloth hinge. A straight piece of cloth glued on, that allows them to fold in half. (You can experiment with sellotape or gaffa tape first for the prototype.)
On one of the semi circles on the same side as the strip, you need to glue a raised strap (again play with tape) that your hand (where your fingers join it) will fit snugly in. This will be reasonably close to the hinge.
On the other semi circle, same side as the hinge, you need to glue a thumb strap. This will probably be further from the hinge than the hand strap was.
You now have the opening closing mouth mechanism of any opening mouth puppet, and can practice.