No. And yes. No, because to legally start a business in Canada as an individual, you need to be a Canadian citizen or a landed immigrant and as a foreign worker here on a work permit in Canada, you are neither.
And of course, as Canada work permits are usually valid only for a specified job, employer, and time period, you would violate the conditions of your work permit by starting a business in Canada. This means that you are not legally allowed to do work "on the side", or even do other work for the employer listed on your work permit.
Yes, because this doesn't mean that you can't apply for immigration to Canada under the Canadian Experience Class of Immigration or apply for business immigration to Canada.
Under the Canadian Experience Class of immigration to Canada, a temporary foreign worker with at least two years of full-time (or equivalent) skilled work experience in Canada or a foreign graduate from a Canadian post-secondary institution with at least one year of full-time (or equivalent) skilled work experience in Canada who has gained his or her experience in Canada with the proper work or study authorization can apply for permanent residence in Canada. And permanent residents in Canada, also known as landed immigrants, are allowed to legally start a business in Canada.
Business immigrants, of course, come to Canada for the express purpose of starting a business. When you look through the information on business immigration to Canada from Citizenship and Immigration Canada, you'll see that there are three types of business immigrant: investors, entrepreneurs and self-employed person.
You might also start a partnership or corporation in Canada (but not a sole proprietorship) by teaming up with one or more Canadian citizens or landed immigrants. See As a Non-Canadian, How Can I Open a Business in Canada? Seek legal counsel about this first, though, before you do it, as it may invalidate your work permit.
For more information on work permits in Canada, see Working Temporarily in Canada (Citizenship and Immigration Canada).
