When was the acronym ACTS first used to refer to components of prayer? When was the mnemonic acronym ACTS, standing for Adoration, Confession, Thanksgiving, and Supplication, first used to refer to the components of prayer?
1883.  The earliest printed example I have found specifically referring to the acronym ACTS is from a serial story by "Marion Harland" (Mary Virginia Terhune) that was printed in an August 8, 1883 publication of the periodical The Continent:


  Our Mr. Burgess once informed a youthful theologue in my hearing that "the monosyllable 'ACTS' formed an excellent epitomical guide in the composition of the principal prayer offered in public worship.  This should begin with Adoration, proceed to Confession, rise into Thanksgiving and close with Supplication."


Mr. Burgess here is the name of the pastor in the story, and apparently not a historical figure.  However, the author's husband was a Presbyterian minister, Edward Payson Terhune, so if the acronym was not original to her, it's possible that she picked it up from him.

Earlier authors referred to the same four elements in the same sequence, but did not mention an acronym.  For example, the Principles of Christian Philosophy (1836) includes:


  Generally speaking, prayer may be arranged under the heads of adoration, confession, thanksgiving, and supplication.


These four components, however, were recognized long before this, but not always in the same order, and sometimes tantalizingly close to an unrelated use of the word "acts."  For example, The Evangelical Magazine, in 1807, printed:


  Many ministers whom I have attended, have seemed forgetful of the nature and design of social worship, so as to fail of introducing those acts of adoration, confession, supplication, and thanksgiving, which constitute the principal parts of prayer.


The online dictionary of Christianese provides some other early examples, but nothing as early as the ones I've provided here.  Still, I wouldn't be surprised if an example earlier than 1883 is out there somewhere.