Yes, you can submit an application for an OWP along with your PR application, the form is here
http://www.cic.gc.ca/english/pdf/kits/forms/IMM1249E.PDF Check box "C" and the fee is $150 for a work permit.
If you submit this request with your application you don't need to extend your visitor visa as well, you will have implied status until they process the OWP. Put it in a separate envelope, mark the outside "OWP - To be processed with Approval in Principle" and put that envelope inside your PR application package.
You're out of work until you have an OWP in hand.
According to the CIC website (
http://www.cic.gc.ca/english/study/study-who.asp) you won't need a study permit for a course of 6 months or less so it will depend on the length of your on-line courses.