Is it possible to be allergic to seaweed?

No. Unfortunatley, you can not eat seaweed because generally it contains such high amounts of iodine that it would make the reaction you have to the substance much more intense.

You might find this medical explaination interesting for an individual lacking iodine:

Seaweeds have a salty taste that is an indication that the material can disperse phlegm accumulation, particularly as it forms soft masses, include goiter, the thyroid swelling that indicates severe iodine deficiency. Thus, seaweed is enriched in with iodine and you can not eat it.