When updating the Assigned to or Additional assignee on a VTB card you get an error "You do not have permissions to modify this record".
When going to the record that card is linked to you can update the assigned to field.
To edit VTB cards a user needs to pass the list_edit ACL for the related field you are trying to change.
The user is most likely failing this ACL check.
Allow the user to pass the list_edit ACL for the field they are trying to change in VTB. In this case, assigned_to or additional_assignee.