There was no purchase of commissions in the militia. The Lord Lieutenant of the County appointed officers and signed the commissions. It appears that only the Adjutant’s commission was signed by the Sovereign.
When not embodied for service, the militia staff consisted of the Colonel (or commandant) and the Adjutant. The militia was subject to call out each year for up to 28 days training. This training could be cancelled by Act of Parliament, for example, in 1820.
Appointments were made to the militia even when disembodied.
I believe that in 1811 the Irish militia was allowed to serve in England and English/Scots militia in Ireland. If locations are noted, the records no doubt held at the National Archives.