Is it important who submits the paper to the journal?
It doesn't really matter who submits it, since it's just an administrative procedure. 

What matters is that the authors and co-authors are properly listed in both the manuscript and the submission system. 

In every submission system I ever used, you have to define who is the main author and who are the co-authors, and there's usually an option to register their contact/email address so they can receive updates on the status of the procedure.