Because you are dealing with so many species, I'm going to recommend that you call the Illinois Department of Natural Resources wildlife biologist who covers that area. His name is Rich Whitton and he can be reached at (618) 993-7148.
Unless the dogs come into direct contact with a sick animal, or one that has very recently died, there shouldn't be concern about them picking up anything from the animals. But, if the animals are being poisoned, you do want to make sure that the dogs aren't consuming any of the carcasses. Rich will have more advice.