On the store itself it now says that it will not be available for purchase on the Xbox 360. This is surprising seeing how the game has been advertised as one of their main games when Xbox had the timed exclusive for the console. The Call Of Duty series has also previously been heavily advertised for that console as well.
This is what it actually says on the site regarding the Season Pass:
"The Call of Duty®: Black Ops III Season Pass delivers 4 epic DLC map packs in 2016 with all-new multiplayer content, including re-imagined fan-favorite maps from Call of Duty®: World at War and the Call of Duty®: Black Ops series, plus thrilling Treyarch Zombies content, all for one great price.* Fans that order the Season Pass will be among the first to experience the return of Zombies with "The Giant" Map in all of its undead glory on Day One. Set in a World War II-era research facility featuring the weapon-upgrading Pack-a-Punch Machine, re-live the chaos of Treyarch's classic "Der Riese" Zombies map, picking up the Zombies story with Dempsey, Nikolai, Richtofen, and Takeo where Origins left off. Season Pass and "The Giant" Zombies bonus map are not available on Xbox 360."
It is unknown if this is just a temporary thing, or if the Season Pass will not be sold for the console at all. We will update this story when we know more.