Just because you got married in Las Vegas doesn't mean that you have to get divorced in Las Vegas. To get married in Las Vegas, of the party needs to a resident of that State for at least 90 days. Since you all have resided in Missouri for 12 years, you are officially a resident of Missouri (congratulations?).
If the marriage is broken (which is obvious by the description) and it can not be repaired then the dissolution is the proper process. A divorce primarily is a method of terminating a marriage, which is essentially, a contract between two individuals. From a legal standpoint, a divorce will give each person the legal right to marry someone else, divide the couple's assets and debts and determine the future care and custody of their children (if you have any).
In order to file for a "Petition for Dissolution of Marriage in Missouri", you or your spouse must meet the strict residency requirements. These requirements permit the court to have jurisdiction of your case, resulting in allowing you to use their judicial system. One party must be a resident of the state of Missouri, or is a member of the armed services who has been stationed in Missouri for 90 days.
You or your spouse must file a petition with the county you live in because that county will have jurisdiction in your dissolution. In Missouri, the marriage has be irretrievable before the Court can grant the divorce. When you file the petition, you will have a hearing date. When you or your spouse appear at the hearing, one or both of you must present the evidence that the marriage is infinitely broken. This could be documents, witnesses, an affadavit from the both of you, etc.
You could hire a divorce attorney or do it yourself, PRO SE to file the Petition. Here is the information that I found most helpful for being a PRO SE (Self-Representation): http://www.divorcedex.com/divorce/Pro-Se-Divorce-699.shtml
Here's the reference on the Missouri statues for the dissolution of marriage (divorce). http://www.moga.state.mo.us/STATUTES/C452.HTM
I hope this advice helps - thank you for asking and have a great day!