I am an American Citizen in New Zealand. For a Visa do I have to mail the documents to Wellington?

Besides send application by post, you can also submit the application to the China embassy in Wellington in Person or entrust someone else to do it.

In fact, I'm afraid that your application will be denied because you are a tourist a New Zealand. Why don't you apply for a China visa in USA? If you travel all over the world for more than 3 months before entering China, it is possible for you to get a China tourist visa in New Zealand. A single entry tourist visa is valid for 3 months, if you obtain it in USA, then travel in the world for 3 months, the visa expires when you enter China, therefore, you have to obtain a China visa en route.