The THETA V (please confirm you have a V) has a limit of about 1 picture every four seconds. As the camera is new, you may be able to get this lower through trial and error.
As far as I know, the camera cannot take a picture when the camera is processing the equirectangular image (the 4 second delay)
The camera can’t save a picture as dual-fisheye to speed up repeat picture taking (the requirement for your application).
If your application requires pictures taken more than once every 4 seconds, you can try the 4K video and then extract the frames from the video and process it.
It’s better to check for completion and then move the motor. If you’re having problems with that, you can start by putting a delay (example 5 seconds) after you take a picture, then move the stepper motor.
Just so you know, my son builds the applications with a delay instead of checking for completion because it is easier to get it going faster. He’s not building anything for production use. In my opinion, you should start with the delay and get it working because it will be satisfying. Then, if you want, you can then add in error checking to make sure your application still works if for example, you lose WiFi connectivity due to WiFi interference. Make sure you save your code to git or something so you can have a working version of your code as you try to improve it incrementally.
If this is a long-term setup, you will need to disable sleep on the camera and also power the camera.
Eventually, you may want to move the camera to the USB API over time, especially if it is a permanent install. Koen has in THETA S in continuous use for more than a year!