I don't know of any expedite companies that fit the bill for you but if you're just looking for a truckload or LTL carrier you should be able to find something even if the majority of carriers are only looking for cross-border eligible drivers. One company that springs to mind is OK Transportation in Etobicoke. They advertise regularly in Truck News that they only run Canada. I think they do a lot of local work in the Toronto area and also run the Toronto to Montreal corridor.
Manitoulin doesn't run Canada exlusively but I think the vast majority of their freight movements are within Ontario. Just don't expect to go faster than 90km/h (except when you're rolling downhill). Speaking of slow trucks, McKevitt also does a lot of runs between their various Ontario terminals. I think their headquarters is Thunder Bay but they have a terminal in Mississauga.
If you're looking to go to Vancouver or something long haul like that, I'm not sure who you should talk to. Most of the companies running that far want teams and I'm not sure if that's something you're comfortable with. You don't need to have a team partner already. They'll pair you up with someone. Maybe try Challenger, Bison or TransX.
You could also try getting a local (day cab) driving position with a company in your area and then move into long haul once you've been cleared to cross the border.
Have you tried the Driver Link website (driverlink.com)? Fill out your profile on there and mention that you only want to run Canada and see what kind of response you get. I get calls all the time from companies that saw my profile on Driver Link. A lot of them are smaller companies with only a handful of trucks and they can't afford to advertise in the trucking magazines.
Young. So I don't mind plus Canada has a very very nice places to go to. I can't go to the US for a while so Canada only. Local, or long haul! Been looking for over 2 weeks and Maybe came around to 4 companys tho. Who want 5 years exp.