I am a U.S. citizen and need to travel to Shanghai for business. Where to apply for the visa?

Elliott, you should go to the embassy or one of the consulates, which are respectively located in Washington DC, San Francisco, New York, LA, Chicago, and Houston. Which one to go depends on where you live. BTW, you don't have to go in person. You can ask someone else or an agent to submit the application on your behalf.