Internal organs have to be harvested while the patient is still perfusing. The requirement is declaration of brain death after certain protocols are followed in the hospital. However, just because a person is brain dead does not mean their body is dead and a person can be kept alive by artificial means for quite some time. If the person is truly dead on scene, they are not a candidate for internal organ harvest.
Here is the upside to this, not that there is much of one. Tissue, skin, bone, and corneas can all be used post-mortem within a certain time frame. So even if they are dead, if they are fairly fresh dead they can still donate the above bodily items.
Find out who does your organ transplant coordination in your area. It's always good to have the number handy. In some states, like Missouri, they have a special team, and a person will respond to a scene to talk to the grieving family about organ and tissue donation. They are specially trained to deal with this very delicate situation. Not saying a regular EMS provider could not address this with family, but sometimes it's better to use your resources.