Rhode Island follows the federal rules for combat zone service, so pay that is excluded from your federal taxable income is also left out of your Rhode Island taxable income, and the federal filing and payment deadline extensions apply to your state return as well.
There is nothing to apply for. Exclude qualifying combat zone pay from the income you report, and use the same extended deadlines the IRS allows for combat zone service.
Your orders or Leave and Earnings Statements showing your combat zone service dates.