I'm in Ontario. But, I'm also interested in knowing the answer to the OP's question.
I don't think the answer is simply you can't start a business unless you're a Canadian citizen or a permanent resident. There are different types of businesses. Like, sole proprietorship, partnership, and corporation.
As for
sole proprietorship, my understanding is that, it covers
self-employment including independent contractors. I know international students who work as drivers for Uber, UberEats, and SkipTheDishes. They're not employees. They're self-employed independent contractors. I myself worked for Uber as driver when I was student on study permit and filed my taxes last year with no problem.
As for
corporation, my understanding is, to register a corporation in Ontario, at least %25 of the directors - not necessary owners - must be resident Canadian. If the number of directors is 3 of less, at least one director must be resident Canadian.
So, I think you might start as self-employed in e-commerce under your name. Then, when you get the PR, register a corporation.
Visit the page on the link below, and find out the laws related to Quebec.
Sole proprietorship, partnership, corporation or co-operative?