The fact that your undergraduate degree is from an unaccredited institution is a significant hurdle. Most colleges and universities will not accept an unaccredited undergraduate degree for consideration in hiring, regardless of your doctorate. They'll likely require you to have completed at least some graduate-level coursework from accredited institutions. Your accredited doctorate helps, but it doesn't entirely negate the issue of the unaccredited undergraduate degree.
In short, while your doctorate is a necessary condition, it's insufficient alone. You would likely need to address the lack of accreditation from your undergraduate institution through further accredited education or significant professional experience to demonstrate competency equivalent to an accredited undergraduate degree. Contacting specific colleges and universities' human resource departments to inquire about their specific requirements is strongly recommended.