Why not the dealer? Because the lien on the car is owned by Toyota Credit Canada Inc (or Toyota Financial Services - as recently changed) and not Toyota Canada, or the dealer where you bought it.
Because that's not the dealers job to handle your additional payments. Toyota Financial Services is a distinct company from Toyota Canada and it's dealers.
Deal directly with TFS. I'm sure they will gladly take your cheque by mail - and I wouldn't be surprised if they accepted VISA, MasterCard or AMEX (don't quote me). I think you might be even able to authorize them to deduct it from account. But like I said - call the company that OWNS your car rather than SERVICES your car. They are different.
If you call TFS, you'll get the truth in 2 min...or you could drive around town into the dealer to ask a question from the next available person who may not have a flipping clue on how to handle this unusual request. And all the dealer would do is mail the cheque to TFS anyways...so why accept the risk of dealing with the middleman. That's how things get screwed up.