Do I have to see a specialist to get an esophagram ordered?

Not necessarily. If your primary physician suspects a problem with the esophagus, he or she may be inclined to order an "esophagram", which would likely be performed by the radiology department. This test can be ordered by your regular doctor - if there is a specific problem discovered, you may then need to see a specialist for further evaluation and treatment.
Not to order, to do. An esophagram is an xray test, typically done by a radiologist (xray specialist); years ago, ; now, not the case. Some gastro docs did their own when indicated, it an be ordered by a primary doc, or a specialist, e.g. Gastroenterologist, ear nose throat doc.