I would like to adopt my wife's daughter. Her father is no longer living and the daughter is over 21 years old.

I don't really know if "adoption" is an option; she's no longer a minor. She would probably be able to use your last name but I don't know about officially adopting her.

No, in most states in the US, you can not adopt an adult. You can include her in your will, she could go to court and change her last name, but adoption is not really a choice.