What is the likelihood of a 4-year-old child getting a goiter?

Goiter. Uncommon -- but it can happen. Especially in cases where there is a strong history of autoimmune thyroid disease in the immediate family. See a pediatric endocrinologist for help.
Relatlively low. It is possible to have a goiter in a four year old, but not likely. There are quite a few other causes of a neck mass in a child that age, which are more common than goiter. Regardless, . Any neck mass in a child needs to be evaluated by your local ent.