Is there such a thing as varicose veins in the stomach?

Yes, not exactly on the stomach, but rather on esophagus and gastro-esophageal junction. The varicose veins there are being formed due to a condition called "portal hypertension", which is a sequence of hepatic cirrhosis. Due to the changes in the liver caused by alcohol, different forms of hepatitis etc., pressure in the veins draining stomach through the liver is rising, which leads to development of submucosal esophageal and gastro-esophageal varicosities. One of major complications of esophageal varicosities is bleeding. Hope it helps!