US citizens require a visa, and you need to apply for it before going (ie. not available upon arrival).
According to http://www.indianembassy.org.cn/ there is no consulate in Chengdu, but Shanghai/Guangzhou/HK, you should contact them directly to see if you can apply via mail or have to go there in person.
Thanks Rick and Diamond for your feedback, I called Guangzhou and Beijing and they seemed to say that I could submit the application and passport through a friend or agency locally. I guess I’d have to put my passport in the mail, is that a bad idea?