Is it possible to have a script added to the checkout page to be able to properly track conversions within AdWords?

Yes. If you go to your site, to the Settings > Instructions tab area, you will see a number of different site pages listed. Instructions are blocks of text that appear at the beginning of the site page as styled text, but the </> allows for inserting code into the instructions text. There is a specific setting for Checkout Successful Page instructions that will only be shown if there is a successful checkout on your site.