According to OED, both spellings are actually about the same age, formed from the existing word "acknowledge" plus the suffix "-ment". The earliest citation for the spelling with e is 1560, while the earliest without citation for without is 1574 (and both happen to be translations of Calvin's sermons).
He will employe all the residue of the life god hath geuen him to make to God acknowledgement thereof, that he might not be found vnthankful.
A. L. J. Calvin's Serm. Songe Ezechias
Let vs cast downe our selues before the face of our good God, with acknowledgment of our sinnes.
Arthur Golding J. Calvin's Serm. on Job
(It's interesting to note that Arthur Golding, in this book from 1570 (another translation), also spelled it "acknowledgement").
When the original word ends in "-dge" and "-ment" is added, it's pretty common to have two spellings, one with e and one without:
It's interesting to note that "abbreggement", the original spelling of "abridgment" (from the French word) was always spelled with the e. It was not until there was a d (when it was "assimilated to the spelling of abridge" [OED]) that people began to spell it without an e.