The feature you want is usually called tethering or remote camera control. Many recent Canon and Nikon DSLRs support it, and these brands commonly provide APIs/developer tools for controlling the camera from a computer.

If live view on the computer is a requirement, focus on newer DSLR models that explicitly support tethered shooting with live view, because support varies by camera and software.

There isn’t one simple master list of “all compatible cameras,” but in practice the safest choices are:

• recent Canon DSLRs
• recent Nikon DSLRs
• some Canon PowerShot models using CHDK

For Canon PowerShot cameras, CHDK can add experimental PTP-based remote control on many supported models, including tethered shooting and image transfer. However, that route is more hobbyist/experimental than buying a camera with official tethering support.

So the best approach is to start with cameras known to support tethering, then narrow by your other needs such as live view, stills vs. video, and available SDK/API support.