As I remember, the local government stopped providing some supplies to the garrison the week before the firing on Fort Sumter once it became known that Lincoln was sending armed ships south.
Lincoln was smart enough to know his actions meant war. (If not, he shouldn't have been president.) From the New Orleans Daily Picayune of April 11, 1861 quoting from from the New York Herald of April 6:
A Western Congressman, who has satisfied himself as to the purposes of the administration, by interview with the President within the last twenty-four hours remarked that "blood would be spilled in less than ten days."
Long before that Anderson had been sending reports up North detailing his shortage of supplies. The first report from Anderson that Lincoln saw said that unless resupplied he would have no choice but to surrender. Lincoln was acting on information he received from the commander on the scene. He could hardly be expected to do otherwise.
Lincoln was smart enough to know his actions meant war.
As was Jefferson Davis. The difference is that Lincoln's resupply didn't have to result in war while armed attack on Sumter could hardly be expected to lead to anything else.