Yes, you can definitely power the All Access. You will need to power the Mini Amp Gizmo with the All Access' power supply - the one included with the Mini Amp Gizmo is not enough to power both.
As Martin mentioned, the All Access adapter has a 2.5mm inner diameter, and the Mini Amp Gizmo has a 2.1mm jack. You could replace the plug on the All Access adapter, or get a 2.5 to 2.1 adapter cable, but since the Mini Amp Gizmo jack is spring-loaded, it holds on to a 2.5mm plug pretty well.