A good olive oil should be hypoallergenic. (the cheap ones - even extra virgin - are often mixed with soy oil)
It's everything derived from seeds/kernels you should watch out for. Coconut oil and often palm oil have traces of seed storage proteins in it. Those are the guys you really want to avoid. (cross reactivity with gluten)
Olive and avocado oils are made entirely from the fruit, so no storage proteins.
It's not common to be allergic to lard. But again, make sure it's not hydrogenated and from healthy animals. :)