Is it permissible for me to carry out work that I verbally agreed upon with a contractor, and then later learned that he acquired his original project through bribery?

If you are certain that the contractor obtained the privilege of executing the work through bribery, then it is not permissible for you to work with him. This is because doing so would be assisting him in the evil act of bribery, which is one of the gravest sins. The Prophet (peace and blessings be upon him) cursed both the briber and the bribed. If you abandon this for the sake of Allah, then perhaps Allah will compensate you with something better, in accordance with His saying, the Most High: "And whoever fears Allah - He will make for him a way out and will provide for him from where he does not expect."