Just install the camera software on each PC.
Should not be a problem if the camera has a static IP address and multiple PCs should be able to access the camera even at the same time.
If you are using multiple cameras then each camera needs to have a static IP address so the end user can identify the webcam or phone to select.
Get one PC working and capable of taking pictures. First with a webcam and then add phone cameras.
(Not sure about how the phone cameras may integrate with the overall requirements...)
Then apply what you learn from that setup and configuration to get a second PC capable of taking pictures via the camera.
Then PC #3, #4, etc..
Caveat being some limit on the number of allowed camera software...