Both body builders have been around a long time. Thomas was founded in 1916 and has been building buses since 1936. Blue Bird was founded in 1927. For a long time they were independents and bought chassis to suit their customers needs. However all that has changed.
Thomas is now part of (wait for it) Daimler Trucks North America LLC. So no surprise that Thomas uses Freightliner chassis for its type C (conventional) and D (transit style) school buses. Thomas claims that fully one third of the school buses on the road today are Thomas Built. (They are no longer built in Canada, just in the US.)
Blue Bird, no slouches themselves are part of North American Bus Industries. NABI produces Optima and NABI transit buses, and is owned by the Dutch company Traxis Group BV. Blue Bird will build buses on most chassis presumably, but International seems to have the lead by my casual observation. Blue Bird has two factories, both in Georgia, and no longer builds buses in Canada.