Unfortunately there is no general Vegard-like rule of mixtures that I'm aware of. The band gap is a single number that encompasses complex energy dispersion relations in a solid, and the temperature/structure/composition relations are typically non-linear.
The good news is that there is a lot of data out there for common semiconductors, especially for GaAs ,GaN and others of the III-V family. I recommend the excellent data base at the Ioffe institute for many common (and less common) semiconductors