Elementary age has such a large age span.
I agree that finger and hand puppets are something simple for students to begin with. So are sock puppets which are simple hand puppets, just like Lambchop was...

For the youngest students you can make paper bag puppets. These are simple hand puppets, and you only have to decorate the front of the bag with scraps of paper, felt, buttons, etc. This is the simplest construction for students as it's very easy to glue anything to the paper bag.  There are patterns for students to cut out shapes to make recognizable puppets to tell familiar stories. However, you can also let students create their own designs from scratch.

For longer projects for lower elementary students, you could make glove puppets. if you used pre cut felt to do props and symbols from fairy tales, folk tales and invented stories. If you want them to use puppets to tell stories. If you buy the puppets, then the students could spend a day using them to retell and invent new stories.

Also, you can fold a paper plate in half, and you have the start of a mouth for the "big mouth" puppets so many students are familiar with today. Then you can decorate around the top of the pipe plate, just be sure you tie string or a shoe lace inside the mouth to the puppeteers fingers on top and the thumb below. Otherwise, the mouth will stop springing open once the paper becomes used...

Older students can make sock puppets, and they only need one sock, clean, and scraps of felt, buttons, and fabric along with craft glue.