A camera shop can show you digital videocams which have an electronic trigger. They might have a used one at discount.
If you wish you can get one that looks like a professional security camera.
An ordinary consumer style camera will look funny mounted on a wall. If you get one then you'll probably want to hide it behind a silvered window or sphere.
It has to be sensitive enough to record a viewable image for the lighting conditions. This is among your top criteria. Some cameras are not sensitive enough (although improvements have been made since the first models came out).
A digital camera typically shuts off after a few minutes idle time. You must be able to either (a) set it so it will stay on continually so you need only send a 'start record' signal and 'stop record' signal, or (b) install triggers that will turn the camera on, start to record immediately, then stop recording when it receives your 'stop record' signal, then shut it off or let it shut off by itself.
You'll also need a power supply.
Did you have a plan what kind of sensor to use that will detect people opening and closing one of numerous doors? This will be tough. Instead it's easier if you can get access to the switching system that goes to the lock mechanisms.
Or maybe you would rather detect when someone starts pressing the keypad. This will record incidents when someone tries to defeat or vandalize the security system.
Somewhere behind the control panel there may be connectors to let you access the signals you need.
This is where you'll need help from a technician who's familiar with your system.
On the other hand there are other triggers you could use. Audio, changes in light level, motion detector, infrared detector, pressure switch under a floormat, etc.