That was exactly my thought. Even corneas will begin to degrade if the body is not refrigerated relatively quickly. Not to mention, organs need to be matched by blood type and potentially even MHC typing.
In the United States, to transplant a heart, the donor must be brain dead. The transplant surgeons will clamp the aorta and pour ice directly into the chest cavity to make sure there is no warm ischemic time. Liver and kidneys are more tolerant, but there’s no way someone killed in a war zone would go on to be an organ donor. The transplant surgeons would have to be following soldiers around with a centrifuge and a PCR machine to be taking kidneys from the dead.
Now they could be recovering the organs for other purposes. Researchers will buy deceased organs, although reputable ones require the appropriate paperwork.