For my kids I just use a straight green wooden stave about thumb diameter, from a cherry tree (although for a kids bow I'm guessing hazel, sycamore and others would also work well). It needs to have a pretty even diameter.
I strip the bark off most of the bow leaving a band for the handle and some decorative bands top and bottom.
I cut a groove on the ends top and bottom where want the string to sit, then string up with heavy dyneema fishing line. If you like you can leave like this. It'll be better than the toy bows and last at least a season.
If you want to go a step further once the bow has a set curve you can secure it to a board and bend it back the other way to make a recurve shape. Be careful, some will snap if you go to far.
They're great kids bows and look the part. They have enough power to be fun but not enough to be lethal.