I am only citing sources, I cannot confirm their accuracy even though I tried a great deal as it's really hard to find any info or sources to back any of this up. Including Ben supposedly using the sound in Star Wars as where this is quoted on other sites seems to be exactly the same as the IMDB one, as if that is the original source.
There is a potential similar sound if you listen to the above original sound at higher speed (x 1.5), then some of the Ewok scenes at low speed (X 0.75). At the start of the arrow firing it has a similar resonance and sound, but is hard to identify with any degree of accuracy with my basic equipment (youtube/browser).
The difference could be that Ben speeded it up or played around with it, adding more electronic effects (which weren't available 40ish years previous). Or perhaps with Ben being expert in this specific field he could have entirely (re)created it himself.
It could also be that it was a different arrow sound to the one above taken from the original Robin Hood, or of course the IMDB quote could just be wrong.
You could perhaps try contacting Skywalker Sounds (at the bottom) or IMDB and see if they can verify the quote.
Additional trivia I found along the way:
In an interview (written), Ben Burtt stated:
Being a young boy, of course I loved action, you know, Westerns, and
things with explosions in it, and violence, all the things that
interest boys at that age. So what I meant was that when there was
something well done—a great fight scene, with punching sounds, or
gunshots, or airplanes—it was exciting to hear. One of my favorite
films was The Adventures of Robin Hood with Errol Flynn.
And it had the great sound of the arrows in that movie. I went outside
with my own bow and arrow, and it didn’t sound anything like the one
in Robin Hood. And so I got really interested: Well, what was that
sound? My bow and arrow didn’t have this beautiful, almost musical
sound as the arrow flew through the air. And so I began to realize
that the sounds in movies were enhanced in some way, there was
something different about them.