What kind of visa do I need for teaching in China?

Yes, you can try to apply for an L type in the Chinese Embassy in Bangkok. Personally speaking, the agency will help you to do get it.
You are suggested to prepare the following materials in advance: a valid passport and a photocopy of the data page, an application form, a recent-taken color photo, your entry or resident permit in Bangkok, and an invitation letter from the agency. By the way, they had better not mention you will work as a foreign language teacher in China, otherwise you will be required to apply for a Z type and it's much troublesome.