Puppets can be made out of anything. 
Stuffed animals hooked up to strings or cardboard drawings taped to sticks
Hand puppets
Puppets, or stuffed animals of different sizes can be used to give the illusion of distance. 
In the the play The Mango Tree (renamed Don't Touch The Mangoes!) the actors are disguised as larger versions of the puppets. The smaller hand puppets appear through openings in the backdrop,  as if they were far away. 

Instead of trying to hide, puppeteers can blend in the scene. You can dress in the same tones as your backdrop - black on black or white on white - or be a prop. If your puppets are birds, be the tree.
Foldable stage the size of a sheet of paper
4" x 101/2" (10 cm x 29 cm) 
The Mango Tree
Lamma Island, China
String puppets
Disguised as a puppet
This stage, folded, will fit into a manila envelope. 
Cut out puppets are simple to make, 
have children draw the animals.
You can make a simple board game that children can color, and use cutout characters of a story