I recently heard it in the movie Rio Grand (1950) so I know it does not originate with Larry the Cable Guy [born 1963].
In 1836, the Western Monthly Magazine and Literary Journal printed "This was unlucky, for as I had to do all my caulking and carpentering myself, I knew I could not get her done by afternoon, and I had made an engagement with Ned Harris and another smart chap for a sail."
There are also numerous cites in 1865 congressional hearings containing get her done, and they, too, are referring to finishing boats or ships.
I believe the expression comes from calling ships her and morphed into a meaning of her as this here job.
Interesting ngram here, if you care to see it and explore.
Thanks for the ngram example.
The nautical origin does seem likely.