No, it's not necessary to keep a journal for The Master Key System.

The course is written in such a way that each weekly lesson builds upon the next one so you can judge your progress according to whether you are able to do the exercises or not.

That's not to say there would be any harm in keeping a journal, quite the opposite. Journals provide a useful alternative perspective that can give you insights into the kind of person you really are, compared to the kind of person you think you are.

After all, "inner" work is really all about discovering the real You.