I came across the idea of making leather mushrooms here. I have a box of leather scraps (remnants from a furniture factory) that yielded perfect materials, but vinyl would work just as well. These were very easy to make and quick (except for the time needed to allow the glue to dry). I used Gorilla Glue, but I've read that rubber cement also works well for gluing leather as long as little strain is put on the bond. If you've tried using rubber cement on leather, please let us know how well it worked for you in the comment section.
Cut two circles (the same size) for the top and bottom of the mushroom cap. Cut a roughly rectangular shape for the stem.
Cut two slits to form an X in the center of the circle you wish to use for the underside of the mushroom cap.
Lay a bead of glue on the wrong side of top piece of the cap. Lay the top and the bottom of the cap together (right sides out). Make sure that the middle area remains glue-free.
Create the mushroom stem by spreading a thin even layer of glue on the rectangular piece and rolling it up. Hold together with binder clips until completely dry.
Once the glue on the mushroom cap is completely dry, stuff it. I used diy recycled stuffing made from thinly cut strips of plastic grocery bag, but poly fill or wool would work equally well. Stuffing is inserted into the cap through the X-shaped slits you made before gluing the top and bottom of the cap together.
Attach the stem by squeezing glue into the hole in the underside of the cap and pressing the end of the stem in as far as you can. You may need to prop the stem against something to hold it upright as it dries.