The chick extraembryonic membrane for waste storage and gas exchange is:
Allantois evaginates as endodermal diverticulum from embryonic hindgut surrounded by splanchnic mesoderm, protruding into extraembryonic coelom during day three incubation. Its mesoderm becomes highly vascularized via VEGF signaling, fusing with chorion forming chorioallantoic membrane where fetal vessels appose shell pores for respiratory gas exchange. Distal sac accumulates nitrogenous waste as insoluble uric acid since embryo cannot excrete via yolk. Thus single membrane serves dual physiological roles: waste storage and breathing organ, expanding to occupy most coelom, later contributing to umbilical vessels.
Ref: Gilbert, Developmental Biology, 12th ed., Chapter 12: Allantois - waste storage and gas exchange extraembryonic membrane.