During the Credits
There are no extra scenes during the credits of Halo 3: ODST; however, the credits are displayed in the same style of the transmission loading screens shown between levels.
After the Credits (spoilers)
Just like Halo 3, there are two extra scenes after the credits of Halo 3: ODST. After beating the game on any difficulty there is an extra epilogue after the credits that shows Sergeant Major Avery Junior Johnson interrogating Vergil. If you turn the joystick to the left, the camera will turn and reveal Buck in the corner messing with one of the monkeys from the previous Halo games.
Again, like Halo 3, the second extra scene after the credits is only shown after beating the game on the Legendary difficulty setting. The scene shows the Covenant finding the entrance for the portal controls to the Ark.