The stork is a traditional symbol of birth. (Before they are old enough to hear about actual reproduction and sex, children are usually told the stork brings babies to their parents.) The reaper refers to the Grim Reaper, a mythical figure sent to reap souls with his scythe to bring them to the next world after death.
In Cat on a Hot Tin Roof, the themes of birth and death are intertwined. Big Daddy is dying of cancer (though he does not learn this until the end of the play) while his two daughters-in-law are hoping to become pregnant. Big Daddy's son Gooper wants his wife pregnant as often as possible so his large family will endear itself to Big Daddy, making them more liable to benefit financially from his death. On the other hand, Brick, who Big Daddy much prefers (along with Brick's attractive wife, Maggie), is not having children because he is gay and does not want to have sex with Maggie.
Birth is viewed as a way of conquering death, or at least, evening the score. Big Daddy might die, but he hopes his line will continue—preferably through his preferred son and daughter-in-law.