How can I write a thesis if my master's program does not offer it?
While it wouldn't be, formally, a thesis, you can always write and publish your research. That would actually be preferable to a thesis that might get buried in university files. 

But you probably need some guidance in going about it the first time. I'd suggest that you talk with a trusted faculty member to see if you can get advice and feedback if you want to do that. 

But if a thesis-based degree is actually a hard requirement, you may need to study elsewhere, as you probably won't convince your university to change its policies. 

However, your dilemma surprises me, assuming you are now studying in Canada and wish to stay there. I wonder if you have accurate information about the requirements for entry. If you do, I wonder why master's level place would be out of sync with what is required for doctoral education. 

I would understand it better if you are studying outside Canada and want to later study there, but still wonder why they would have such rigid requirements.