Actually, transfusions of cadaveric blood did occur, mostly in Russia, in the 30s, and the results of such transfusions helped in developing the anticoagulants and preservatives that are now present in donated blood. It never really caught on though, and live donors are much preferred.
Harvested donor organs are usually transfused within a matter of hours of being collected, and are kept in O2 perfusion fluids. Some harvested tissue can be stored, but not organs.
Live donors can be repeat donors, you can screen them for infectious diseases and other transfusion hazards on an ongoing basis, and more importantly there are more live people out there to fulfil demand!
The current number of registered donors in the UK is around 1.4 million - you would need an awful lot of dead people to make any sort of significant contribution :)
Go on, do something amazing - become a blood donor!