The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.
The warning on the cars.gov site says: "This application provides access to the DoT CARS system. When logged on to the CARS system, your computer is considered a Federal computer system and is the property of the U.S. Government. Any or all uses of this system and all files on this system may be intercepted, monitored, recorded, copied, audited, inspected, and disclosed to authorized CARS, Dot, and law enforcement personnel, as well as authorized officials of other agencies, both domestic and foreign."
Are you willing to turn in your car and the ENTIRE CONTENTS OF YOUR COMPUTER FOREVER in order to get your Cash for Clunkers Rebate?
Although it certainly sounds like it should be a violation of your 4th Amendment rights, apparently it is legal for them to do this!