Secession declared the State of Arkansas independent of the United States; it didn't change its form of government. After secession most Southern states drew up new constitutions, which usually included provisions for organization of the state military.

Never heard of any county keeping militia records. The militia was organized on a county basis by the state government. Commissions of county militia officers were signed by the governor, giving them lawful authority, just like all county officials, from judges down to constables and justices of the peace. Militia officers were elected locally but reported to the state adjutant general and submitted muster rolls to him. Orders could also be given by the governor, since he was actually in command of the militia.

You would do well to study Arkansas state laws from that period dealing with the militia. County commissioners took steps to care for the needs of families of Confederate soldiers or their widows, but unless state law gave them authority to regulate the militia, I'd expect to find those records in Little Rock.