This picture shows the development inside a bird egg:
This shows the connection of the embryo to the yolk sac. Does this mean birds have navels? And if so, what happens to the umbilical cord once the bird hatches?
All animals develop in this way, whether they are oviparous (developing in an egg) or viviparous (developing inside their mother, or live-birth).
From Wild Birds Unlimited:
As for their umbilical cord, it looks like it falls off when the chick hatches if it is not fully absorbed along with the yolk sac, along with everything else contained in the egg (although there is not a lot of material left within the egg by the time the chick is ready to hatch).