According to CNN, pardoned turkeys go to a farm to live the rest of their time. "George Washington's Mount Vernon estate [has] a sizable wooden pen built specifically to house the [...] turkeys".
ABC News reports President George W. Bush's turkeys went to Kidwell Farm. The turkeys are put on display for the public, but generally die within months.
Unfortunately, the CNN article and others go on to explain that the turkeys selected to receive Presidential pardons were bred to be eaten; by the time they are pardoned, they are grossly overweight and physically unfit for extended survival. Historically, most have died from illness within a year of being pardoned.
I would assume that due to their lack of longevity, unless you were one of the qualifying farms selected by the President or his staff to house and display the turkeys, it is unlikely you would be able to adopt one.