As far as I know, all battery powered busking amplifiers are designed to run corded, dynamic microphones and dynamic in this case stands for passive. The alternative being condenser microphones which need a power source (often but not always 48v phantom power).
I use a Behringer XM8500 dynamic microphone which performs very much like a Shure SM58 at a third of the price and of course a microphone cable and run it into a AC-33. I DO also use a condenser instrument microphone system for my ukuleles but its expensive.
Headset microphone systems are condenser microphones that need power sources and while they can be successfully used they will be spendy.
Anthony