The parents can be served through publication if their whereabouts are unknown. Even if they respond, after more than a year without contact or support their consent to the adoption is unnecessary. Good luck.
For informational purposes only; not intended to, and does not, constitute legal advice or a legal opinion.
Yes, Ohio law allows non-parent adoptions even when you don't know the current addresses of the parents. In fact, it makes it somewhat easier since you don't run the risk of them showing up and contesting the adoption. Even if they did find out, their consent is not needed if they haven't seen him or helped support him for over a year.
There are several ways for you to proceed. However you definately need the assistance of a good adoption or family law attorney in Ohio. You could start off by getting temporary guardianship - that is usually faster than adoption. At least with guardianship you will be able to establish that you can make decisions for the child and it will also help establish the abandonment by his parents.
Most attorneys have become very adapt to finding information and people through the internet. Often what people would not say publically, they say on facebook. Your attorney will have a variety of ways to try to locate them. At the end of the day, if they cannot be found, the court will allow you to move on to adopt them.
Your great nephew is a lucky little boy to have someone care so much for him. Good Luck