On the child's menu, there is a choice between chicken or shrimp, with grilled veggies and steamed rice. There is also so much food served, that you could easily share whatever you are having to supplement your child's meal if they don't love chicken or shrimp.
Everything is brought to the table/grill fresh and all seasonings and oils are added as they are cooked. Typically peanut oil is the oil used for cooking because of it's high smoking point, and sesame oil is added later if at all for flavor. I don't know for sure what Teppanyaki uses. I don't know if there is a substitute available for the high heat cooking at the restaurant.
From my understanding, WDW is very accommodating for food allergies/special requests, but these need to be noted on your ADRs. You should be noting your child's allergy with every ADR that you make. The chef will visit you and in some cases will call you before you visit to discuss meeting your needs. I would mention your concern with the CM when you make your ADR and see if there is a way for you to speak with the restaurant now to discuss if this is the best choice for you and your family.
Happy planning.