Follow this link to the Wyoming Vital Records website. They have some of the forms and information to help you get started. Please note that the Report of Adoption form and the Adoption Application/Request for Birth Certificate Following Adoption form are not enough to start the adoption process! You must file these two forms along with a Petition for Adoption! There is no official Wyoming Petition for Adoption form, so should talk to an attorney about writing the Petition. 

Wyoming Statutes and Court Rules

The Adoption Statutes: Title 1 - Code of Civil Procedure Chapter 22 - Adoption