Byzantium continued to trade after Rome proper fell. Moslems disrupted the sea trade between the West and India in the 7th Century. It also disrupted much of the Mediterrean trade and helped accelerate the coming of the Dark Ages in Europe.
The 540AD 'Dark Ages' were a worldwide event...recorded in the tree-rings all around the world.
By then Rome had actually moved to Byzantium. The Empire was already much reduced and wasn't trading much at all.