... shooting things, automatically.

The shooter's challenge is to break it, at range, using a shotgun.

Yes. I've seen some usually excellent shooters trying to throw their shotgun after a series of misses. But I digress. :)

At present, my basic approach is to use two independently movable turrets, one for the sensor package, and the other to operate the shotgun.

Why two turrets ? Mount the camera on the shotgun. After all, that's how it works if the shooter is human.

I have a vague idea of how I'd go about it - a low-res color camera with some kind of processor attached to it looking for orange blobs for initial acquisition, and perhaps some variant of lidar for precise tracking - but I've no idea if these are correct, practical, or within the reach of a self-funded hobbyist. So, I ask you for your thoughts.

The camera is probably enough. After all, the clay pigeon is fairly slow and fairly close, and the cloud of shot is a fairly large (i.e. not point-sized) affair. You'd need the precision tracking if you're trying to shoot clay targets at several hundred meters with a rifle.

You can probable gauge the distance of the target from its size and/or from where it appears.

Then you need a bunch of mathematics to come up with a firing solution. There are several ways to tackle this - some examples would be using a plain-vanilla PID controller or solving the equations of motion in real-time (I've done something similar a while ago, while writing an automated player for the classic arcade game Asteroids. It was only for a 2D case, but adding the third dimension shouldn't be a problem).

Then you'll need another bunch of mathematics to actually rotate the shotgun into the right position and fire. Ideally, the time to do this is included in the algorithm mentioned in the previous paragraph, so you have to calculate the firing solution only once and can then rotate the shotgun into position without having to keep checking whether your solution is still valid.

Personally, I'd be content with simulating the whole process. I'm a bit wary of computers that control guns. Even though I'm a control engineer.