Have there ever been any "Lebron James" of academia?
Stephen Wolfram self-published a physics book at age 12 and then 3 more by the age of 14 (a lot of people do this, but you can look at the PDFs by clicking on the links here and I think you'll find that these could compete in quality to the works of many adults at the time).

That same link will tell you that he did not finish high school, or undergrad, but received a PhD at age 20 from Caltech. On the recommendation of Richard Feynman and Gerald Freund, he joined Caltech's faculty at age 21.  


  Have there ever been any academic researchers who showed so much
  promise at one stage (in undergrad or grad school maybe) that they
  were offered funding to set up their own research institution, or a
  high-ranking position they'd typically be unqualified for?


To qualify into the PhD program at Caltech, one typically needs at least an undergrad degree, and Stephen Wolfram did not have one. As user StrongBad points out, entering a PhD program at Caltech is far more exceptional than entering the NBA straight from high school, when you look at the percentage of people that have done either. Entering the faculty at a university like Caltech at age 21 and only 1 year after finishing the PhD, was also probably less common percentage-wise, than entering the NBA straight from high school.

You ask about "setting up their own research institution", at first I was going to tell you that including this in your question was a bit extreme, but after reading more about Wolfram I found out that he set up the Center for Complex Systems Research (CCSR). 

However LeBron James stayed in the NBA for a fairly long career and has continued to be (arguably) the best player in the NBA. Wolfram "academic career" was short relative to most other academics, because he shifted to doing other things such as: 
(1) making software for academics
(2) making educational resources and doing outreach
(3) managing businesses that help academics
He did continue to do academic work and published a book with academically original content, but it wasn't received as well as the mid- to late-career achievements of LeBron James. His success in (1), (2), and (3) though can definitely be compared to the mid- to late career achievements of LeBron James.