By run your website, I assume you mean host it in Canada, but only offer shipping throughout the USA?
The answer is yes, in fact you can host your website in whatever country you like, including the US, and designate where you will ship to. If you are only wanting to ship in the US, I would suggest that having a US domain would save some confusion rather than having a Canadian domain and not shipping to Canada....would be messy!
If it's your own site, you make your own rules : )