This type of flannel board is not a board but rather a big pillow.
Rectangular reading wedge pillows make great flannel story pillows, as do bed rest back pillows. The firmer and taller the back rest, the better the flannel board. Also, it is better to find one that does not have arms. While it may be expensive to purchase a back rest pillow new, if you already have one or have access to a used one, then it makes a unique and versatile flannel board.
Large foam blocks work well, if they can be found at a reasonable price.
Once you know the size and shape of your flannel story pillow, purchase enough flannel or felt to cover. Measure each side of your pillow. If you have a cube that is 2 feet x 2 feet x 2 feet, purchase more than 12 square feet to compensate for seams. For sewing novices, you will need to learn to sew a seam and understand how to sew a pillow.
Finish this project one of two ways; stuff the pillow in an almost-finished pillowcase and hand stitch the last seam shut. For experienced seamstresses, sewing in a zipper makes for easy on and off for washing.