Not even if my contract starts on September 1st? I was hoping to get my vaccine in Canada as soon as my quarantine ended.
If your work permit is approved by August 26th, then you can fly to Canada. Otherwise you will not be able to board the plane due to the current travel restrictions in place. The fact your contract starts September 1st is irrelevant - this doesn't allow you to travel. You need to get the work permit approved if you want to travel anytime before September 7th.
As of September 7th the travel restrictions are changing. From September 7th forward, you can travel on an eTA but ONLY if you are fully vaccinated. If you are not fully vaccinated, then you need to wait for the work permit to be approved before you will be able to travel to Canada.
If you are not yet fully vaccinated, I would fix that asap. The rules in Canada are evolving quickly. It's possible that at some point soon those who hold work permits may also need to be fully vaccinated to travel to Canada. Also, many workplaces (e.g. schools / universities, banks, government, etc.) are also now starting to implement vaccine mandates for their staff.
EDIT: As of now, processing times for work permits from France are estimated at six weeks. So that will give you a rough idea of what timeline to expect.