You can only use the Vital Records office as a Wyoming background check resource if you are doing research on a family member, or the record you want is more than 50 years old. They are strictly confidential otherwise. If this applies to your situation, you can make a request for a fee and possibly wait for 2 weeks to get papers back.
Criminal History Records
This is a more restricted Wyoming background check resource and is usually only utilized by employers who do employment screening. You have to submit a completed fingerprint card in order to get a history record back so it is not something you can do casually or online. Each search will cost $15. Additional information may be found through the court system instead. Try the Wyoming Appellate Case Management System for online searches based on case numbers.