yes. Many jurisdictions have rights in a mark that arise simply from use. That said, it is not the same as the rights flowing from registration and in some jurisdictions, it is not available.

A quick call to a trademark lawyer should allow you to clear up if the rights you get through use are sufficient for your purposes.

Note: Branding and brand strategy is the most overlooked aspect to starting a business and a little expense now can save you a whole lot of problems later on.