I cannot speak for the Church, but I have seen webcams for the Nauvoo Illinois, San Bernardino California, Rexburg Idaho, and Twin Falls Idaho Temples. Most updated every 30 seconds but could be set (with the browser) for longer update times. The Nauvoo Temple included an archive feature where you could go back and look at stored images in one-minute intervals from the time of groundbreaking until dedication.
The Nauvoo site was hosted by Deseret Book. The Rexburg site was hosted by BYU-Idaho. I believe the others were done privately.
It was (and is) a delight to see them go up.
I am not a lawyer, but I believe that anyone with the resources would be free to set up a webcam if they had the permission of the camera location owner. Countries and some local jurisdictions do have different privacy rules, so I would be careful there. And as a member, I would certainly want to be sure that the site was of good quality.
The Twin Falls Idaho site is active. You might be able to check with that site owner and the San Bernardino site to see how they were able to work out the details.