Any CNC router of adequate size would do what you want. Then you would need to add some kind of 3D scanning hardware/software in order to locate the features on the existing door. 3D scanning systems are available from relatively inexpensive to ''you can't afford it''
If there are undercut features on the workpiece, then you would need more than a 3 axis machine, maybe 4 or 5 axis.
You could build the machine you need or just buy a new or used machine. My personal preference is fixed gantry and a moving table. Much more stable machines, but moving gantry machines work fine and take up a bit less space.