No. Tom Cruise is not part of the movie "Bring It On".
Bring It On was directed by Peyton Reed and written by Jessica Bendinger. The movie starred Kirsten Dunst, Eliza Dushku and Jesse Bradford. See full cast and crew at IMDb.
The 2000 movie tells the story of competing high school cheerleading squads. A champion high school cheerleading squad discovers its previous captain stole all their best routines from an inner-city school and must scramble to compete at this year's championships.
See What the Cast of "Bring It On" Looks Like Now at Cosmopolitan.