I chose to make the vests out of heavy duty craft felt, mostly because it doesn't fray at all, but it also happened to be ridiculously cheap (about $2.50/yard when on sale!). Fleece would also have been a great option, but I've made costumes out of it before and it's a bit harder to trace on darker colors, and it's thicker and stretches a bit, so it's harder to cut accurately. I've seen people just use black T-shirts too, and cut the arms off.
The felt is super wide at 72" so I kept it folded in half and cut out 2 layers at a time for both the front and the back. I was lucky to have a leftover Halloween costume pattern that had a vest template for sizes from 3-8. I tried one on scrap fabric and made a few modifications for a better fit. I made 2 different sizes for the little kids and big kids (even the Kratt brothers are different sizes!), and got 16 vests out of 4 yards of felt, which was about $10. I should add that I did then take the front and trim it into a V-neck to match the original design.
They didn't have to right color felt for the appliqués, so I went with flannel instead. I fused the flannel to a paper backed fusible web all in one piece, then traced and cut out the different size circles. I used a rotary cutter to make quick work of cutting out all the squares and rectangles. After that, just peel off the paper and iron them all down! I just eyeballed the placement of the circles, but a small ruler helped space out the squares pretty evenly and easily.
I really, really should have just stopped there. But I know how to sew and I tend to pay attention to detail, so I actually went and edged stitched every single piece of flannel down. Completely and totally unnecessary, and virtually unnoticeable since I used matching thread. But it looks darned good so it was worth the extra effort to me.
Now's when I sewed the front and back together. Again, I could have stopped there, but I added some edge stitching all around the openings, thinking it would give the felt some added stability. In fact, they were even able to stand upright!
Voila! We purchased some $1 magic gloves at the craft store to complete the look, and the kids all had costumes for the party, and a pretty decent party favor to take home!