A little searching of my own. References cited.
The word sharpshooter stems from the Old English (before 830) word 'scharp' appearing in the book 'Ancrene Riwle.' It meant pretty much what it means today; cutting, keen, or sharp. It's a relative of the Old Frisian 'skerp' also meaning sharp. The modern Dutch word is 'scherp' the Old High German 'scarf'...modern German 'scharf'.
Our word 'sharp' was first attached to the word 'shooter' (sharpshooter) in 1802. It's meaning, of course, one who is keenly accurate with a gun.
It’s a story that’s sometimes told and you can understand why, as a connection between sharp and Sharps seems obvious. It has also been said that the term was popularised during the American Civil War of the 1860s. Wrong war, wrong country, wrong rifle. The stimulus was the Napoleonic Wars and the term is British. So the short and sharp answer is, no, there’s no truth in it.
Doubters may like the facts. The Sharps rifle was designed by Christian Sharps in the late 1840s and made from 1850 onwards by his firm, the Sharps Rifle Manufacturing Company. But the term sharp shooter had been in use in Britain since no later than 1801. The Experimental Group of Riflemen had been set up in the British army in 1800; this led to the creation of the 95th (Rifle) Regiment in 1802 as a specialist sharpshooting force using the Baker rifle. If you’re familiar with Bernard Cornwell’s books about Richard Sharpe of the 95th Rifles, then you will already know about this period and milieu.
I found the term in the Edinburgh Advertiser for 23 June 1801, in an item on the North British Militia: “This Regiment has several Field Pieces, and two companies of Sharp Shooters, which are very necessary in the modern Stile of War.” It quickly became common, appearing in the Times more than 20 times in the next three years. In 1805, a report could say baldly in the expectation of being immediately understood that “Lord Nelson was wounded by a French Sharpshooter.”
Bavarian and Austrian riflemen and sharpshooters are recorded earlier. The Tirailleurs (French for sharpshooters) were Austrians who fought on the French side early in the Napoleonic Wars. The German term Scharfschütze for them is recorded in Jacobsson’s Technologisches Wörterbuch of 1781, so it seems certain that the term was borrowed into English from German as what linguists call a calque or loan translation, in which each element of the word is translated literally.