It depends! Definitely don't give oral vitamin E unless your child's doctor prescribes it for a specific medical condition, i.e., a deficiency. Using vitamin E oil on a small area of the skin should not cause problems, but it may not be the optimal treatment for what your child has. Your child's doctor can tell you what is best.
Yes. Unless the child has broken skin or allergies to ingredients in the oil, it should be safe.
Olive oil. More than just salad dressing! vitamin E oil is a common home remedy for skin problems and scar healing but, alas, there's not much info on whether or not it really helps. Also, vitamin E can cause bad skin reactions in some folks. Consider using just plain olive oil to moisturize your skin [if that's what you're trying to do]. That would be safer in kids. Good luck!