Can childbirth cause anemia?  

Anemia. Yes. The fetus during pregnancy draws many substances from the mother, one of which is iron. If the mother has a poor intake of iron, then she is prone to iron deficience anemia. Also if the mother has a large amount of blood loss during delivery, this can cause anemia.
Yes. The developing baby draws a lot of nutrients, including iron. Most women gain water weight during pregnancy, which lowers the blood concentration. A fair amount of blood is lost during the labor process. These all contribute to anemia of pregnancy. Check with your doctor to make sure you are getting enough nutrients in your diet.