We built our home a year ago. We want to finish our basement using the plans provided by our builder. If it is just building the interior walls, could we qualify for a permit to do the work ourselves following their plans?

As an owner of the structure, you can do any work on your own home if you live at the home. If you have anyone else work on it, they must be a licensed contractor.  A permit needs to be obtained.  Having those original plans should help you in submitting for the permit but the plans would need to be updated to show how you are proposing to finish it.