The Wikipedia page does mention the wolf and moose populations in Isle Royale National Park as its sole real-world example. The paper it cites, though, doesn't seem to find the LV equation to be among the most useful available models, which is a pretty bad sign for its actual descriptive power.

I've got a book somewhere (small trade paperback, dull silver cover[1], title might be Life) which claims that no one has ever gotten those pretty predator-prey equations to cycle nicely in the real world, not even with two species of micro-organisms in a test tube.

With 3 species, the LK equation can become chaotic, so I wouldn't expect to be able to duplicate a real-world history even if the model were perfect.

Perhaps we could find a 2-species real-world LK case involving bacteria deep underground.

This source claims that some real life examples have actually done this correctly including the archetypal rabbit/lynx example.

Upvoted for your footnote :)