According to the April 2008 edition of the Harper's Index in Harpers magazine, Radiohead has won the exact same number of Grammys as Barack Obama—2.
Best Alternative Music Album, 1998: OK Computer
Best Alternative Music Album, 2001: Kid A (shared with producer Nigel Godrich)
Best Spoken Word Album, 2006: Dreams of My Father
Best Spoken Word Album, 2008: The Audacity of Hope
These numbers don't take into account certain Radiohead-related Grammy awards, such as the one for Best Recording package won in 2002 by Thom Torke and Stanley Donwood for Amnesiac.
And yes, it looks like "Grammys" is the proper plural.