Is it possible to only teach K-3rd grade English?

Yes, it's possible to only teach K-3rd grade English. Many elementary schools have a self-contained classroom model where one teacher is responsible for all subjects for a single grade level. In other schools, teachers might specialize in a specific subject area within those grades, such as English Language Arts (ELA). So a teacher could have a position dedicated solely to teaching English to students in kindergarten through third grade.