We are experiencing the same error.
When a user with Home, Orders (not edit) and Products permissions clicks on the "Mark as fulfilled" button on the order detail page they are sent to a page titled "The order you're looking for couldn't be found".
However, when a user with full access clicks on the same page, there is no error and the order can be fulfilled.
This issue first appeared on July 29th. Prior to this, the limited-access user could fulfill no problem.
I added permission to Draft Orders and our user with lower permission is now able to mark as fulfilled!
Our user previously had permission to Orders, Edit Orders, and Settings only and was able to mark as fulfilled with no problem until about a week or two ago.