Al Gore may not have won an Oscar himself, but he was a big part of it, wasn’t he? Is he the first guy to win (well, be involved with the winning party) an Oscar AND a Nobel?
John Steinbeck won a Nobel Prize in Literature, and the screenplay of his book The Grapes of Wrath won an Oscar. Obviously, Steinbeck’s writing was a big part of that Oscar win.
George Bernard Shaw was instrumental in what became My Fair Lady.
Oh… never got that much publicity? Well, ignoramuses like me haven’t heard of him. Thanks. 
G.B. Shaw was credited (among others) for the screenplay for Pygmalion (1938). It won “Best Writing, Screenplay” that year. In fact, his imdb.com trivia page says
That discounts Steinbeck because The Grapes of Wrath (1940)'s Oscars weren’t for writing (though it was nominated for it), I guess. Though by the OP’s broader definition, he qualifies.
According to the IMDB, GoW won a best screenplay Oscar, though it went to Nunnally Johnson. Still, it’s within the parameters as given.
Near miss: Harold Pinter. Two Oscar nominations in addition to his Nobel.
Gore also won an Emmy for his Current TV network.
Now if we can just get him a Grammy and a Tony he’ll be as big a star as Rita Moreno.
Al’s hoping for a Golden Globe. That’s the one with the real prestige.
[Latin accent]
But not as esmokin’, papi!
[/Latin accent]
John Forbes Nash won a Nobel for Economics.
Though he had precious little to do directly with the Best Picture-Winning A Beautiful Mind, the movie was about him, and he did provide some consulting, so if you stretch the parameters, then he’s in.
Did Gore really win an Oscar? Was he the producer of that film?
Just looked it up. As I thought, Gore did not win that award. Davis Guggenheim did