The default casualty-carrying vehicle, the ambulance hasn't evolved much during the GTA series. Best place to find em? The hospital, of course.
If multiple people are killed an ambulance will arrive and two paramedics will get out and attempt to revive the wounded. Typically if paramedics are killed it creates more police attention than other civilians.
'Paramedic' missions can be toggled on or off when in the ambulance in all GTA III-era games. These require the player to find, pick up and deliver patients to the hospital within a time limit. With every level comes another patient to deliver, increasing the difficulty. Usually when the player beats level 12 the mission is complete (but can be carried on and replayed later) and a reward is achieved, eg. infinate sprint or extra health.
If the ambulance is left during the mission the player has failed and must restart from level one, similairly if they kill the patient or run out of time. Three patients can be carried in an ambulance at any one time.