Supposedly the word digicam (digital camera) originated in 1989 as a trade name in England, but I have not found any further information.
The Free Dictionary offers the following definition of digicam:
digicam n. Informal A digital camera.
However, the first occurrences of the word digicam in a Google Books search results use the term to refer to a TV videocamera for shooting videotape.
Here is the Ngram chart for digicam for the period 1970–2005:
The three earliest instances of digicam (short for "digital videocamera") that a Google Books search finds are from 1980—and all three refer to videocameras manufactured by Ampex (of Redwood City, California) that debuted in 1979. From Filmmakers, volume 14 (1980) [combined snippets]:
Internal circuitry compares the camera's current operating conditions with a previously established set of operating parameters, and automatically adjusts the camera to optimum settings. Ikegami and Hitachi offer computer set-up as options on their top-of-the-line studio cameras. Ampex offers the BCC-14 Digicam, which is a computer controlled ENG/EFP camera. While a computer set-up camera is expensive to purchase initially, time and manpower savings make such a system attractive to high volume production facilities.
From an Ampex advertisement in Broadcasting Cable Yearbook (1980) [combined snippets]:
THE BCC-20 DIGICAM.
Our new EFP camera features computer-in-the-head technology that combines a microprocessor and digital memory with a powerful software program for full-time video control of the camera.
Digicam can achieve 0.06% registration in all zones of the TV picture quickly in addition to giving the operator automatic or manual set-up with results that surpass those of a conventional camera.
And in an Ampex advertisement from International Broadcast Engineer, volumes 170–190 (1980[?]) [combined snippets], cited by Josh61 in a comment above:
Built by Ampex in our new California facilities, the Digicam was designed to produce the highest quality picture both on location and in the studio A Computer In a Portable Head. The BCC-20 Digicam's computer-in-the-head technology combines a microprocessor and digital memory with a powerful software program to give you full-time digital control of the camera.
Even with the power turned off the Digicam's memory-on-the-head stores the operating parameters.
The TV Camera Museum confirms that the Ampex BCC-14 ("Portable Colour," according to the site) was released circa 1980, and the Ampex BCC-20 ("Portable ENG Colour") in 1980. Various Ampex Digicams are mentioned in publications throughout the early 1980s.
Though the drop-off in frequency of use of the term digicam since 2003 (indicated in the Ngram graph above) may be exaggerated by incomplete data, I suspect that, as cell phones have become more and more capable of handling video and photography, the consumer market for digital cameras and videocameras (sometimes called camcorders) has waned. For its part, Ampex went out of business in October 2014, according to the Wikipedia article about it.