It's hard to prove a negative, but I suspect that the story is bogus.
Knoblock, The American clipper ship, 1845-1920: A comprehensive history, p. 106, has this:
Because of the special conditions prevailing aboard the clippers, crew difficulties and mutinies were seemingly on the rise in the 1850s. Historians Howe and Matthews record in their history at least sixteen instances of mutiny, crew difficulties, or murder (there were likely many more that thay do not mention) that transpired between a crew and the ship's officers or captain from 1851 to 1861 aboard the following clippers [...]
He then lists the following mutinies, plus several more that he considers "noteworthy," and describes some of the mutinies in some detail, continuing through p. 112.
- Challenge 1851
- N.B. Palmer 1852
- Aurora, Sovereign of the Seas 1854
- Undaunted 1854
- Atalanta 1855
- Ocean Express 1855
- John Milton 1857
- Morning STar 1857
- Black Prince 1858
- Tornado 1858
- Adelaide 1858
- Golden State 1859
- Messenger 1859
- Stag Houng 1860
- Boston Light 1861
- White Swallow 1865
- Dashing Wave 1869
- Snow Squall 1858
The boom years of the clipper ship era began in 1843 as a result of a growing demand for a more rapid delivery of tea from China. It continued under the stimulating influence of the discovery of gold in California and Australia in 1848 and 1851, and ended with the opening of the Suez Canal in 1869.
So neither "early" nor "late" 19th century seems especially likely. Knoblock's book seems to cover the entire time period when the ships existed, but he's only focusing on American ships, and we don't know whether the ship in Heinlein's anecdote was American.
A possible source that Heinlein might have read was Clark, The Clipper Ship Era, An Epitome of Famous American and British Clipper Ships, 1910. Clark discusses, for example, the Challenge mutiny on pp. 181-188. But searching the text and index for "mutiny" didn't turn up much. Another source that Heinlein could have read is La Grange, Clipper ships of America and Great Britain, 1936. I didn't turn up much of interest with keyword searches.
My suspicion is that the story is bogus. Although mutinies were apparently pretty common, the ones with the more lurid details, such as the Challenge mutiny, pop up in every source. The story about the teenager taking command and sailing the ship home is so flashy that it seems hard to believe that it wouldn't be discussed in sources like Knoblock, Clark, and La Grange.