Yes. We have added this feature in the latest version of Notezilla . Download it from here: https://www.conceptworld.com/Notezilla/Download
Although it is not automatic, which means the item will not move down as soon as you mark it as completed. You can still do it in just two clicks.
Right-click inside the Notezilla note having the checklist and choose More->Sort-> Checklist.
This will shift the items marked as completed, to the bottom of the list.
Here's a short video on this feature:
Creating a checklist inside a sticky note that uses the Standard editor:
Creating a checklist inside a sticky note that uses the Markdown editor: