Try without the single quotes.
[staff-directory orderby=sort_order]
You can further order them by
[staff-directory orderby=sort_order order=asc]
or
[staff-directory orderby=sort_order order=desc]
The help section indicates “the attribute to use for ordering. Supported values are ‘name’ and ‘ID’. (Ex: [staff-directory orderby=name])” so I am not sure if you can use custom field. If this is the case, and you only have one category, you can sort them by “order=asc”
If sorting is paramount and you have multiple categories, I found for now just change the date and time of each staff–very similar to changing date and time on posts. This orders the way I’d like to see it (until the plug-in has built in ordering system).