Military pay you earn while stationed in Rhode Island is not subject to Rhode Island income tax if you are not a Rhode Island resident. Pay from a separate civilian job you hold in Rhode Island is still taxed by the state.
There is nothing to apply for. Leave your military pay off the Rhode Island nonresident return, and report any other Rhode Island income on it. The Division of Taxation nonresident instructions explain how.
Your orders showing you are stationed in Rhode Island, proof of your home state of residence, and your Leave and Earnings Statements.