Animals DO get CRPS. There are various reported case studies of CRPS in animals, just off the top of my head, I can recall one case report of CRPS in a cow.

There are also animal models in use for CRPS research. They tend to be difficult to generate because of the current thinking that CRPS is not one single condition and therefore any animal model is limited in its application. Currently the animal models are generally used to research trauma or immunity. In the trauma category, immobilisation is one of the specific features being researched (amongst others obviously).