paperless trail
n. A sequence of electronic files or transactions that document the actions of a person or organization.
Consider the mini-crisis I dealt with a few weeks ago when one Disney executive let me know — by e-mail, of course — that he was furious at another executive because of an e-mail that the other executive had sent him. I followed the paperless trail to its source and read the original offending e-mail.
—Michael D. Eisner, “Shunning E-Mail's Darker Aspects,” Vital Speeches, July 15, 2001
1994 (earliest)
Who knows what 'smoking guns' are lurking in the recesses of computer systems? … Further complicating matters, Johnson-Laird said, is the easy transferability of computer files by such means as e-mail. Other paperless trails may be found in computer-stored or generated records, voice mail and electronic bulletin boards.
—Richard C. Reuben, “Shadow World: Searching information highway's side roads for evidence,” American Bar Association Journal, April 01, 1994
As near as I can figure, the phrase paperless trail has been used since about 1989 as a pun (on the well-known phrase paper trail) that refers to the general idea of working with electronic files (that is, without paper). In this sense, it's a close cousin of paperless office, a term that entered the lexicon around 1971.

For the more accurate sense of a documentary sequence of electronic files or transactions, the earliest use I could find is from 1994.