There is not many instances, perhaps none, even up to the very last day, of a confederate unit completely running out of ammunition. Not so for rations...
There is 1 battle that comes to mind..
The Last Trench
by Louis Beam
General Early says that the Union Army was a primary source of tactical munitions, but he also supports your point. Rations of course were another matter. The Confederates were all too often forced to live off the land and plundered Union stores. BTW, I got carried away on this thread. Nathan Bedford Forrest was an incredibly strong and inspiring tactical leader and a great strategist. Given more support and better commanders, western operations could well have gone the Confederate way with Forrest as the cutting edge.
He clearly seemed to have a better appreciation of the biog picture than the commanders under which he served. E.G., Had Hood permitted him to go after Schofield as he was withdrawing from Franklin, the Battle of Nashville could have gone the other way ... maybe. Hood delayed him too long, and he was beaten back.
Happy Birthday to him!