Is it possible to be allergic to iodized salt? If you were, would it be the iodine or some other ingredient that you'd react to?

No. Although some people may get a pimple-like skin rash if taking in too much iodine but it will take a ton of salt for you to get to that stage. There no such thing as iodine allergy per se and it is not related to seafoods.
No. Iodine ion and of course the sodium and chloride themselves are all essential molecules of life. If somebody claims they are "allergic", they are spending too much time reading internet disinformation. Iodized salt has saved millions of people from lifelong mental retardation (i use the term without apology) and death. "iodine allergy" is to iodides, not iodine itself.