You get the same six-month extension to file and pay Oregon income tax that the IRS allows for service in a designated combat zone or hazardous duty area, and your spouse gets it too if you file jointly. You also get 180 days from the day you redeploy from the combat zone or qualified hazardous duty area, or from the day you are released from hospital treatment for injuries or illness you got serving there.
There is nothing to apply for in advance. File your Oregon return within the extended period and note your combat zone or hazardous duty service. Forms are in the Oregon Department of Revenue forms and publications library.
Your orders or Leave and Earnings Statements showing the qualifying service dates, or your hospital discharge documentation.