Is it true that because metal was scarce, the Oscars given out during World War II were made of wood?

No, that is not true.

The only Oscar made of wood was given to Edgar Bergen in 1938. From 1942 until the end of the War, the award statuettes given out were made of plaster, and the recipients received 'real' ones after the war as replacements.

http://www.reelclassics.com/Articles/General/oscar-trivia-article.htm

The plaster Oscars were, of course, part of the war-time propaganda campaign; there was metal enough to make them. When my mother was in high school, her family assiduously scrimped and saved and, to some degree, suffered, in support of the war and of the older son who had been drafted. Meanwhile, my father was in the US Navy and saw the Navy dump canned food into Boston Harbor because they had too much food and more metal than they knew what to do with.
http://www.straightdope.com/columns/read/2395/were-wwii-scrap-drives-just-a-ploy-to-boost-morale

Bergen's statuette was wooden not because of a lack of metal, but to celebrate his wooden dummy. "In fact Bergen was the recipient of the only wooden Oscar statuette ever awarded in history. The special Honorary Oscar was bestowed upon Bergen and his wooden comedy partner in 1938 for his creation of the character Charlie McCarthy."
http://www.talkingcomedy.com/SI-Vent-2005/movies-siVENT05/EBMT-MOVIES-siVENT05.html

The answer is: No, not wood. During World War II there was a shortage of metal and, therefore, the Oscars were made of plaster. Subsequently, the recipients of these Oscars were able to trade them for the "real" thing.

http://www.answers.com/topic/oscar?cat=entertainment