This is very good question.. Venison was the most important game animal in Colonial and Early America. They were plentiful throughout North America. Deer, because of intense hunting, became scare. By the mid-19th century venison had become rare in Eastern markets in America. By the 1900s it was scare in Western states too. By the early 20th century there was the virtual elimination of venison from the American diet.
In the 19th century if they purchased or killed deer, it was very expensive. The haunch was the favorite piece of meat. It was roasted, boiled and fried or cut into slices (collops) and prepared. Most 19th century cookbooks had a few recipes for venison. Venison was served as a luxury in upper class homes and in fashionable restaurants in the late 19th century.
By the early 20th century there were laws protecting deer and other game . In the South, where the deer were more numerous, it still remained on the menu.
In the 20th century as the deer population grew, more was eaten. There were more recipes for deer in 20th century cookbooks.
From: "Oxford Companion of American Food and Drink" by Andrew F. Smith.
Was deer or venison eaten by soldiers during the Civil War?
A good article on this is from "Deer Jerky and the Confederate Soldiers"
see:
https://prezi.com/wa3d4axfnbxm/deer-jerky-and-the-confederate-soldiers/
It states "deer jerky played a big part in the Confederate nutrition. They had to make do with what they could find."
Another reference I found of eating deer during Civil War by soldiers is in book "The Taylor's Civil War" (historical fiction) by Lowell F. Volk. This is about Virginia farm family. There is reference to Union raiders taking everything they had and killing deer for meat.