Can cooking oils made from food allergens such as peanuts and soybeans trigger an allergic reaction?

It isn't likely, though it is best to err on the conservative side and select a different oil, such as corn or canola.  As a result of refining, most oils consist almost entirely of fat and contain no significant protein, the component responsible for food allergies. Therefore, it is thought that those allergic to, say, peanuts can likely tolerate peanut oil. Yet health authorities such as the Canadian Food Inspection Agency advise those with food allergies to not only avoid the product in question, but also derivatives of the product.  
 
Additionally, some oils, especially those described as extruded, cold pressed, or expelled, can contain trace amounts of protein and may pose a small risk to those allergic to the protein. 
 
 Source(s):

Everydayhealth.com "Allergens and Cooking Oils " 
Canadian Food Inspection Agency "Soy--One of the Nine Most Common Food Allergens"