How do you file or can you file for abandonment if the biological father of the two children has not paid child support for 5 years(4 years were paid forward at time of divorce due to the fear of father disappearing) and has not fulfilled/or contacted the children for visitation rights for 8-9 years? I am the biological mother and the children live with me and have since the divorce 9 years ago.

It depends upon where you reside. In many States there is no such thing as abandonment unless the children are basically left unattended at a bus stop. Failure to support and/or visit is not abandonment.

He disappeared?

You can go back to Court and try to force him to support the children. You cannot force him to be interested or a good father.

What do you really want? Do you want to collect child support? Unless you are in GA you can't file for that. Most states have agencies to help collect support.

Do you want to force the father to be a father? Can't do that, since the courts won't go along. They can dictate a visitation schedule, but they can't force him to actually visit.

So, again, what is it you want?