What is the legitimate Islamic (Sharia) position on profiting from a company that has obtained government concessions through illicit means, knowing that I will participate in signing checks that will be disbursed to parties within government agencies?

There is no objection to working as an intermediary for a company in exchange for a known fee (a fixed amount or shares), provided that the intermediation is in something permissible by Islamic law. As for intermediation in bribery, it is not permissible, for what is taken for it is ill-gotten gain (suht), and the Messenger of Allah, peace and blessings be upon him, cursed the briber, the bribed, and the intermediary. It is obligatory not to sign a check that is disbursed as a bribe to a government official.