The answer to your question is, yes, the ULN2003AN will work for this application.
However, I'm curious, do you want to use the ULN2003AN because it is a through hole device and it's easier for you to wire? Because that's what you have 'on hand' (following WIHOH design methodology)?
The reason I ask is that a simple FET will work for this application.
You don't need any power from the Moteino for the ULN2003. The device is just a transistor driver that requires no additional power. Tie any available GPIO to one of the ULN2003 input pins. Connect Moteino GND to ULN2003 and 24V DC Ground. Connect negative terminal of your VERY LOUD alarm to the corresponding output of the ULN2003. Finally, connect positive terminal of your VERY LOUD alarm to +24VDC and do this while wearing earprotectors.
Set the Motenio GPIO with the following code:
#define VERYLOUD_ALARM_PIN x
...
digitalWrite(VERYLOUD_ALARM_PIN,LOW); // do this BEFORE setting pinmode
pinMode(VERYLOUD_ALARM_PIN,OUTOUT);
...
// WAKE UP Neighborhood!
digitalWrite(VERYLOUD_ALARM_PIN,HIGH);
