I am the father of my child and have not seen the mom for almost 10 years. My wife and I would like to have my wife adopt my child. She wants to be her legal guardian along with me.
Unfortunately, the only way your wife can adopt your son is if his mother agrees to voluntarily terminate her parental rights. If not, you would need to consult with an attorney in your state who can tell you how to have her rights involuntarily terminated. Since the IAC only does voluntary adoptions I do not have any information on what is involved in terminating parental rights involuntarily.
If your son's mother agrees to voluntarily relinquish her parental rights you would need to hire an adoption agency or attorney to take the relinquishment (also called a surrender or consent). In addition, there would need to be a home study done for the second parent adoption, where a social worker determined that your home was a safe physical and emotional space for a child. The home study includes a criminal background check. Unfortunately, the adoption process can be expensive, though it is hard to say exactly how much it would cost.