I covered a piece of cardboard from a furniture box (so it's a bit thicker, but not much thicker than a fruit and vege box) with the felt. The piece I had just fit so it is taped on the edges and the back with white masking tape. It seems to be holding well.
You don't even need to be very good at cutting things out. I just did it free hand, and as you can see from the dog/donkey/horse "animal", I'm not very good! It's a great activity for E to do on her own and for us to do together. It's probably not a good idea for me to let her do it by herself though, you may be able to see the tail on the "animal" is missing.... She pulled it off!
When she is a little bit older she might like to make a picture. But for now, she is quite happy slapping those shapes onto the felt board.