Lab tests show I have a mustard allergy. Would that be to the plant or the sauce?

Probably the sauce. Mustard allergy is very uncommon, and very easy to self diagnose. It doesn't take most people very long to figure out they're having symptoms every time they eat something with mustard. Blood tests are notorious for giving false positive and false negative results, and are used far too often. The best way to identify and treat food-related symptoms is to consult a board-certified allergist.
Plant. This exceedingly rare hypersensitivity may be due to mustard seed which is used to make this condiment. Look on the label; mustard usually has just two or three ingredients. Some specialty mustards may contain other ingredients, additives or turmeric. The exact type of test you had and the details of the reaction may determine the validity of the test.