I've just bought a small slab of 8mm copper plate from M-Machine. They didn't have it in stock but found a supplier who had some and ordered it in. They list 3/16" and 1/4" copper, albeit POA. Give them a call.
When I was looking I found a number of commercial stockholders that had 3/16" and 1/4" plate, but almost none had the 5/16" plus plate I needed. You could also try Milton Keynes Metals, they list 5mm copper sheet.
Andrew