I work for a family law attorney and it is possible to have a biological father's rights terminated in certain cases.....x number of dollars behind on child support, no contact for x amount of time, as well as some additional factors. Even if the biological father wants to fight the adoption, he'd have a tougher time trying to prove why the adoption should not be granted if he were behind, hadn't had contact, etc. It's not an easy process but it is possible.
I only watched bits and pieces of the latest episodes but I believe that I am remembering correctly that Robyn stated that their dad was behind on child support and only talked with the kids sporadically (not sure if this is truly the case or not).
It's very possible that the Browns, or at least Kody and Robyn, have met with an attorney who has given them advise on what needs to happen prior to and, if and when, Kody wants to try to adopt the children. We have had clients come in prior to an intra-family adoption, custody cases and even divorce to make sure that they have things lined out before they take any legal action. If that were the case in the Brown's situation, then they would have known that the law would most likely require that Kody and Robyn be married in order for the adoption process to go more smoothly. Also, in our State, a home study has to happen during the adoption process which would mean that the State would assign a social worker (or the equivalent) to go to the home, talk with the adoptive parents, other family members, make note of the living situation, etc. (Sorry for using etc. a lot but there's simply too much to type out). While Kody doesn't live in one home all of the time, it would look better if he were actually married to Robyn when the home study takes place.
**edited to add after barkley's post: They also do criminal background checks in our State too. The process isn't an easy one, that's for sure. But, having your ducks in a row does help.