Trace the image onto the sheet and then paint it. Cut the sheet into puzzle pieces. You may need to sand the edges to prevent splinters. Remember, the saw blade will cut away some of the sheet so you will lose bits of the image. You can also do this with foam core.
Foam core is easier to work with.
You can cut it with a sharp artist's knife. It is also more expensive and damages more easily. This is a good option if not many people will be playing with the puzzle and you won't want to use it again after the festival. To make the puzzle more difficult, paint both sides the same. Or paint a different image on the back to add variety.