Is there any software that can generate the electronic DOS of GaAs?

The $ce{GaAs}$ is one of the must studied semiconductors. One of its main application was as the active site of semiconductor laser, becoming the first stable material used on this devices (by stable, I mean long time duration at room temperature) and it is used in any red (due to the gap) laser pointer.

For electronic data, you can consult this page at the Russian Ioffe institute.

Answering your questions:

• Codes for calculate the density of states (DOS): SIESTA, Quantum Espresso, ABINIT, ORCA, CASTEP (all free), VASP, CRYSTAL ($$$). (almost any software capable to work with periodic systems)*.
• Has DOS of GaAs been studied previously: Yes. In the Ioffe page you will find information about the electronic properties.

*Be aware that one of the pitfalls of DFT is the calculation of the gap for semiconductor materials. But, the shape of the bands and DOS are calculated right.