adoption, adopt, adopted

Definition:

The terms "adopt" and "adoption" refer to the process of someone legally becoming the child of people who are not his biological parents.

The Bible uses "adoption" and "adopt" in a figurative way to describe how God causes people to be part of his family, making them his spiritual sons and daughters.
As adopted children, God makes believers to be co-heirs with Jesus Christ, giving them all of the privileges of sons and daughters of God.

This term could be translated with a term that the language of translation uses to describe this special parent-child relationship. Make sure it is understood that this has a figurative or spiritual meaning.
The phrase "experience adoption as sons" could be translated as "be adopted by God as his children" or "become God's (spiritual) children."
To "wait for the adoption of sons" could be translated as "look forward to becoming God's children" or "wait expectantly for God to receive as children."
The phrase "adopt them" could be translated as "receive them as his own children" or "make them his own (spiritual) children."