Can I use a 9v, 2A AC Adapter with a 12v, 2.5A device?

Yes...

but don't expect the device to work as well as it should,

and in particular don't expect its battery, if it uses one, to be fully charged.

Not enough voltage or current. If the device is a light, it might be dim. Electronic devices probably won't work at all.

AC adapter is used for a device that has the rectifier circuit inside it...usually bridge diodes and filter capacitor..

ordinary DC adapter has rectifier and filter capacitor inside itself ..

WARNIG ..AC adapter may explode capacitor(s) in some devices ..

{like a sound bomb!) or may damage the ICs

Ac adapter is only a transformer