We study vanishing theorems of tautological bundles in the sense of Berget--Eur--Spink--Tseng restricted to wonderful varieties. As an application, we prove a characteristic-independent analogue of Brieskorn's result on cohomology of arrangement complements, in addition to a comparison theorem between Orlik--Solomon algebra and the logarithmic de Rham cohomology of wonderful varieties. In a different direction, we extend a vanishing theorem of Borel--Weil--Bott type for tautological bundles. Finally, we reduce the weak version of White's basis conjecture to a problem about cohomology vanishing of tautological bundles.