When testing for celiac disease do they run a gluten test?

Blood test. Usually doctors will start with getting a history. There is more to gluten sensitivity than just "stomach problems". Then they might order a blood test: anti-tissue transglutaminase antibodies, and/or anti-endomysial antibodies. Depending on those results, they may or may not need a colonoscopy/endoscopy.
No. There are not looking for gluten. Celiac disease occurs when your immune system reacts to gluten in your diet by forming antibodies that attack the intestinal tissues as if they were germs. These antibodies can be found on a blood test (as long as you have been consuming some gluten). A positive blood test can be confirmed as CD with a biopsy showing injury to your gut.