You are mistaken. YES, you can vote in the primaries. There are open and closed primaries. In open primaries independents can choose any party ballot. A registered party member can only vote in the primary.Can't vote in a primary...Independent.
In a presidential primary it is up to the party to either open or close the voting. In the last presidential primary both the democratic and republican primaries were closed which is customary in Arizona.
YES, independents can vote in the upcoming primary