Paramedics do get called by the Police/Sheriffs on calls like you mentioned. When Paramedics arrive on scene they have to perform several steps to decide to start CPR or pronounce death.
If the decision it to pronounce the patient they don't have to make base station contact (Rampart).
All they need to do is document their findings, and if not already present request that the Police/Sheriff respond. They will then remain on scene until they arrive.
If the death is appears to be caused by natural causes, and the patients doctor is willing to sign the death certificate then the body is released to a mortuary.
If death is caused by homicide, suicide or accident, as it appears it will in your story,
then Vince will call Detectives and the Coroner
to complete the investigation and remove the body.
Hope this helps!