Most people would agree with you that a child deserves to know both parents. Most judges will also agree to that. Very rarely do judges nowadays deny all contact with either biological parent. Ohio has a presumption of joint custody, which means that both parents have contact with the child. So the law is on your side and so is the standard, but that doesn't guarantee that you won't get a judge with an agenda.
Yes, it can be court-ordered that they meet you half-way. They can also force the mother to establish some form of communication.
It sounds as if you do not have a court order establishing custody. You need to get one. Yes, you CAN do this without a lawyer, but you would need to be pretty savvy about how to proceed or at least have a paralegal drawing up paperwork. I do not advise doing this without a lawyer. I am trying to advise you to proceed with a lawyer if you can and without one only if you must. Don't give up on the child for lack of money.
Yes, you can work things out without a court order but I strongly advise that you do not do that. What you CAN do (if you can re-establish communication) is to come to an agreement BEFORE going to court. Then the judge is virtually certain to make the agreement official and legally binding.
The trouble with not having a court order is that you will be fighting just to maintain contact for the rest of the childhood. It sounds as if you and your H are reasonable and that the mother is not. That makes it tough to go through this mess without the authority of the court to back you up. This mother sounds like an alienator who will try to exclude you from the child's life. You will need a court order to enforce your stepchild's needs for both families.
If she agrees to what you want, or anything close to it, you could start that up unoffically but do take it to court to get an official court order once you have established yourselves as a part of the child's life.
Another good approach if that fails is to request a mediation. Judges just about never deny a mediation. The mediation gives you a way to force the mother to communicate with you in a meaningful way, with a professional mediator present to keep things fair and balanced (hopefully). If you can come to an agreement during mediation, the mediator will type it up and get signatures, then you take it to the judge to formalize as a court order.
If you do not come to an agreement in mediation, at least you have tried. The judge will not be impressed by a parent who is refusing to communicate and who has hidden a child from their own parent for years.