How to find out if my mother is divorced

If your father filed for divorce, it may have been filed in the county court where your parents had been living at the time when your father left.  If you go to the court and look in the court's computer for either your mother's name under respondent or your father's name under petitioner, you can obtain the case number (assuming that your father filed a petition for dissolution of marriage).  Once you have the case number, give the case number to the court clerk and ask to see the file.  A file that old is probably in storage and not in the court files.  If the court clerk obtains the file, you can't remove the file from the court, but can ask the court clerk to photocopy it.
If your father did not file for divorce, your mother can still file for divorce.  She will need to have your father served by publication since his current whereabouts are unknown.  Service by publication is running a notice in the legal notices section of the newspaper for a specific period of time.  The court clerk can tell you how long the notice has to run to be effective service by publication because that time requirement varies from state to state.  Service by publication is effective even if your father never sees the notice in the newspaper.