Lee was facing an army, and was able to gather quite a bit of stores locally despite that fact.
Then why didn't he press on after Chancellorsville?
He did. He went to Gettysburg. Within a couple of weeks of Chancellorsville he was already advancing northward. Brandy Station took place on June 9th.
He had no supply line established.
Indeed he did. So well established, in fact, that his troops had time to capture and send to Virginia black Pennsylvanians as "contraband." If he had the resources to kidnap people and ship them South, surely he could have used those same resources to send wounded men along the same route.
Union stores where?
The Allegheny Arsenal (still being restocked), Harpers Ferry (already worked once) and the Washington Arsenal.
“Union stores Where? The Allegany Arsenal was at Pittsburg, 150 miles away over the Allegany Mountains,
The Harpers Ferry Arsenal had been in Confederate hand since September 1862 all of its machinery had been moved to Richmond and was no longer in the arms manufacturing business. The Washington Arsenal is a possibility, but only if General Lee could bust through the fortified lines around DC. Unlikely because if for no other reason, he had expended most of his artillery ammunition at Gettysburg.