Yes. The first breakfast cereal product (that didn’t need to be soaked overnight) was the Kellogg brothers’ “Sanitas” or “Kellogg’s Sterilized Bran.” Released in 1895, it was packaged in a plain cardboard box, no bag involved. They developed several additional products that were packaged this way, and spun off (from their sanitarium business) the Battle Creek Toasted Corn Flake Company in 1906.

Later, William Kellogg — the non-physician and arguably less crazy brother who advanced the cereal product and brought it mainstream — differentiated his boxes from those of emerging competitors by putting the boxes inside a sealed Waxtite (waxed paper) bag. Branding and copy (W. Kellogg also pioneered cereal prizes and giveaways) were printed on the exterior bag.

It was William’s son, John L. Kellogg, who developed the now standard practice of putting the bag inside the box. Plastic began to replace wax paper in the late 1950s.

So to address your question about cartoons: Unless you were watching very old material from the 30s, cereal boxes would have had bags of some sort at the time. Why animators choose not to represent them is a question for them — although in a universe where a house cat can order a bazooka by mail, I’m inclined to forgive the omission.