‘Great oaks from little acorns grow’ was first used in that form around the start of the 19th century. Like many proverbs though the thought it contains had been expressed using different wording for many centuries before that.
The first use of the proverb in its precise present form that I know of is from Philadelphia newspaper The Examiner, September 1833:
‘Great oaks from little acorns grow,’ and I hope and trust that from your small beginning, this ancient and honorable Commonwealth will again renew her standing.
That is preceded by an example which is alike enough to be considered the same phrase, in David Everett’s Caleb Bingham – Columbian Orator, circa 1795:
Large streams from little fountains flow,
Tall oaks from little acorns grow.
There have been several variants of the same idea put into print earlier, right the way back to Geoffrey Chaucer’s Troilus and Criseyde, circa 1385:
An ok cometh of a litel spir [shoot]
See also: the List of Proverbs.