Safflower and sunflower oils are the most allergen friendly, start with those.
When you start adding in oils, be careful as many contain ingredient without telling you. Olive oil can have corn. So, how to test olive oil? Put it in the refrigerator, if it hardens, it's the good stuff, if not, you have a cheap knock off. Now, you may not have a reaction, but if you do, you wouldn't want to eliminate all olive oils if it's only because you ate the Walmart brand olive oil.