It's important to be clear as to what a food allergy is.
An allergy is an immediate (or prompt - within a minutes) reaction to a substance, which causes symptoms like skin rash, swelling of lips or tongue/mouth, wheezing or difficulty breathing, diarrhoea, watery eyes or nose, etc. Irregular bowel habit, abdominal cramps, constipation, nausea, bloating, etc. are not symptoms of allergy.
There are a series of well recognised and understood food intolerances - e.g. lactose intolerance, where some people lack the ability to digest milk sugar, causing it to ferment and produce gas and bloating, etc. Coeliac disease is another common intolerance to gluten - a non-allergic reaction to gluten (found in wheat and other cereals) causes the intestine to become inflamed and work poorly.
Many people report symptoms that don't fall into these definitions. These non-specific conditions are often loosely called 'food intolerances', and lumped in together with 'irritable bowel syndrome'. These conditions are poorly understood - in particular, no one knows how or why they occur.
In terms of allergies - there are recognised tests for them, although the tests are not extremely accurate. They include skin prick testing, where extracts of various substances are applied to the skin, and then a pin prick made at each spot, injecting a tiny amount of the substance into the skin. A few days later the skin is examined - the formation of a blister suggests an allergic type response to that substance. There are also blood tests available (so-called RAST tests), again they aren't fantastically accurate - but they can support a diagnosis of genuine allergy. RASTs are expensive, and a seperate test must be run for each substance suspected - before a doctor orders such a test they will normally attempt to narrow down the list with you before ordering only 2 or 3 tests.
Coeliac is easily, and relatively accurately, diagnosed on blood tests - but the most reliable test is an endoscopy to take a biopsy sample of the intestine.
For other intolerances, nothing has been shown to work other than food-elimination diets. People have tried all sorts of blood tests (and there are a number of commercial labs who offer such tests) but the scientific evidence is that they are less accurate than elminiation diets, and much more expensive - few doctors believe there is any role for blood tests in diagnosing 'intolerance', expect to test for diseases which may have similar symptoms.