I second what Dan Anderson said, but want to focus into a specific part of his answer, prayer.
He is very faithful, and has demonstraited it. Thus you don't want to make this a battle of faith. I assume you don't practice his religion, I'd guess your an athiest, but that doesn't matter. This is what is important to him right now and you need him to feel your respecting those beliefs. If you don't do this then your setting him in a position of breaking is faith, and likely in his mind his loyalty to his mother, for you; that is not going to work well.
Instead I suggest you can make it clear that while you respect his reason and faith you don't think this is what God would want from him. God doesn't want him to be unhappy, he doesn't want to force an undue burden on the children he loves, and ultimately your son has to be able to speak to share his faith with other's to help it spread. After all just because the boy promised God he was willing to keep his vow, and demonstrated that willigness, that doesn't mean God demanded or expected him to keep it. In addition the bible (or no doubt whichever religious text is applicable to his religion) shows numerous examples of God respecting those willing to act in Gods name but also freeing them from the commitment. Abraham's willingness to sacrifice Issac is the most obvious example to me, and it's something believed by christian, Jewish, and Muslums, if your son isn't any of these I still suspect that whatever religion he is has some equivelent story. The point is that God is often Willing to not hold his people to harsh acts in his name or to release them from vows. Your son has already shown his faith to God in keeping silent this long, surly it has been long enough now that God would be willing to release him of that vow, would in fact wan thim to be able to talk and live his life.
Of course your a mere mortal, you can't tell him that, but God can. God is the one person who can give him permission to break this vow without feeling he has condemned his faith. That's where the prayer comes in. Explain this to him and ask him to pray to God on it. He can ask God if this is what God wants, when/if God wants him to speak. He doesn't need to pray for permission to speak, only for guidance in what God would want from him. Does God still demand this of him?
If you phrase things this way you give him an 'out', a way to speak without needing to actually break his vow. If in his prayer he does not feel as if God is telling him he needs to continue this vow then that means he is free to speak while still having demonstrated his faith, and love of his mother. Until he feels that time is right your respect him and work with his not speaking, but at the same time you can also make it clear that you would much prefer his being able to speak and that you can't wait until he feels he has been released from that promise. This respects his wishes while still allowing you to reitterate all the reasons life would be easier for him to speak.
If you pray with him you can pray out loud and ask god for permission for him to speak by listing all the reasons why you want it for him and that your son being freed to speak would be a blessing for you because of how much you love your son. That breaking of the vow would not being asked by a selfish son that doesn't want to keep his promise, but by a loving mother who respects her sons faith but also wants loves her son so much she can't stand not having him to talk with. Breaking the vow is now partially a sign of love not of giving up.
I would suggest you seek out a religious figure, as already suggested, also, but I suggest you do this after praying with your son as I mentioned above. In fact pray with your son the night before you go to a religious figure for guidance. Ask God to show her son the right way, and to be there with you tomorrow when you go seek guidance from his priest/rabbi/Mullah/whatever. This means the religious figure will be someone you already prayed for guidance to, someone who God can speak through to give absolution of the vow. Then when/if the figure agrees that God doesn't want your son to keep the vow any longer, that your son has already done more then enough to demonstrate his faith, your have received the guidance you prayed for; litterally as close as you can get to Word of God.
Having said all this, religion may not be the only reason for your son to keep his vow. He may feel this is something he owes his mother, that to speak with you is to somehow be forsaking her. He may not have communicated this to you, either because he can't using just yes/no answers, or because he himself doesn't fuly understand his feelings. Thus in addition to addressing the religious angle I suggest you also address his feelings towards his Mother. Help provide him other ways to remember and respect her, such as making him a locket with a photo of her in it, or starting a routine of thanking his mother for entrusting him to you (prossibly as part of your praying with your son).
If you research, or ask another question on here, your get many other examples of things to do to help him feel close to his mother. For now I just wanted to point out that his feelings may be as much about not wanting his mother to feel he has given up on her, or replaced her with you, and as such you should work to address those potential emotional feelings as well as the religious side of things. Maybe explain you would like to hear more about his mother, who your so thankful for since you wouldn't have him otherwise, and you would love to hear him tell you about her in his own words.