What kind of tests can an allergist/immunologist perform to check for DNA disorders?

Different ones. It depends on the clinical symptoms, age of the patient, and which dna disorder they are considering. There are many different disorders which can be diagnosed genetically (by looking at dna), and there are also specific dna structural disorders. Depending on what your allergist/immunologist is looking for, they can order the proper test, which may or may not be a dna one.