Engineering the Release Mechanism
Most modellers will opt for the easy option of using a servo or some sort of contrived cam to operate the shutter. This has the advantage that the camera is not compromised and it is cheap, but everything else is on the down-side - it is fiddly, adds space and weight to the set-up, and is potentially much less reliable. A typical compact digital will not have a socket for a remote conection - they don't even have viewfinders these days - and although some have the ability to operate via usb commands the complexity of doing this without a PC is significant to put it mildly.
The only real option apart from the servo one is to disassemble the camera, find the shutter-release contacts and make connections to them. This can be a bit daunting, and will obviously spell the demise of the camera for normal use.
You have to get the thing apart, locate the three shutter contacts and solder wires to them, find a way of bringing the wiring out and then get it all back together again. There is not a lot of space inside so this can be a challenge. I removed the shutter button and actuating rod, then found the switch solder pads or the copper tracks to them (scrape off the solder-resist lacquer) and made the connections having first planned the exit route of the wiring through the shutter button aperture.
The contacts in my cameras are arranged so that the focus and release lines are shorted in sequence to a third common line. The first two can usually be tied together and shorted to the third to focus and fire the shutter as one action. However, this may not be universal.