Is There a Toy Box Mode in Toy Story 3 (Xbox 360)?

No, Toy Story 3 on Xbox 360 does not have a "Toy Box Mode." That mode is exclusive to Toy Story 3: The Video Game on PlayStation 3, Wii, and PC (and later Disney Infinity). The Xbox 360 version features the standard story campaign and challenge modes, but not the open-world Toy Box.

If you want Toy Box Mode, you need the PS3, Wii, or PC version. The Xbox 360 port omitted this feature.