Well, three years later, I can answer my own question.
While there isn’t any (that I can find) applescript-related documentation, some is provided for the js rendition at OmniAutomation - Outline. I’d looked at it once upon a time but couldn’t make anything out of it. Seeing this thread got me wondering again, so I grabbed the sample doc and tried again.
tell application "OmniOutliner"
set d1 to document 1
tell d1
organize leaves by {column "State", column "Buys"} under d1 with prune
expandAll
end tell
end tell
Putting the two columns in a list structure like that was the key. For no reason I can discern, I get a “Can’t get document 1 …” error if I don’t do the sleight of hand to send it under ‘d1’. But otherwise, it organises the document and after a sort, the output looks just like in the example.