So-- if you're adding the teacher as a student, make sure the course is published-- the email invites will not show until after publishing the course.
As far as having a Canvas account before adding users to the course, that would depend on your Canvas instance setup. I'm not familiar with what options the free version of Canvas allows, and with a paid version, you should reach out to your Canvas administrator to see how your system is configured.
You might also have the option of sharing the course as a "public" course which anyone can access via a link without a Canvas account, however, participants are not enrolled or tracked in the course.