Can you start out with a pediatric oncology fellow and work your way up to a doctor?

See below. Being a pediatric oncology fellow means you have completed medical school (M.D.) and finished residency training (ave. 3 yrs). A fellow is a "full fledged" physician receiving a specialty training in oncology.
See answer. A pediatric oncology fellow is a doctor that is training to be a pediatric oncology specialist. The fellows are trained by the specialists in academic teaching children's hospitals. When the fellows are caring for a child cancer patient, the fellow is being supervised by a board certified pediatric hematologist-oncologist. In my experience the fellow and teaching physician work together as a team.
You have it backward. You don't get to be a fellow before you become a doctor. You can follow one around and work with them in some capacity, but it would never put you ahead of the line to get into or through medical school.