We haven't managed to get our hands on a copy yet, but David Remnick's excellent review in this week's New Yorker relayed the anecdote:

"Two white aides objected, saying that having the First Lady appear in Vogue, inevitably dressed in expensive designer clothing, would look unfeeling when so many people were living in misery. Two black advisers, Valerie Jarrett and Desiree Rogers, argued that, on the contrary, having an educated, attractive African-American First Lady on the cover of Vogue could be a source of inspiration, and counteract a plenitude of negative images. In the end, Obama posed for the magazine wearing clothes from both a young American designer she helped discover, Jason Wu, and J. Crew."

As for whether Vogue editor Anna Wintour's fundraising for the Obamas played a role in her decision, we'll have to wait for the book.