I've made a few kids bows out of plum branches, ocean spray shoots, and red oak boards. The big box hardware stores have 1/4" x 1.5" x 4ft red oak boards for around five bucks, and it's relatively easy to find straight grained ones. I cut 'em on the bandsaw using a pyramid bow limb template and do zero thickness tillering. You can add a wood riser block at the handle. It's easy to steam in recurves if you want, and for questionable-grained boards, I use TBIII to glue on a brown paper grocery bag backing.
For arrows, I use 1/4" dowels for shafts, with turkey feather offcuts for fletching. Hex nuts or brass threaded inserts make good blunt points threaded onto the end of the shaft. On one set of arrows, I used black rubber stoppers for blunt points, to make them safer. I wanted to test just how safe they were, so I had my wife shoot me square in the back with one out of a 14lb kids bow. It left a welt and was surprisingly painful.
Don't shoot each other with bows and arrows, kids!