I think you'll find that the ideal target spots for a deer will be slightly different that for a coyote. Although they're both quadripeds, the details of anatomy will differ. At the end of the day though the idea is the same.
To get a quick, clean kill you need to disrupt the central nervous system such that functions essential for life are shut down, or you need to cause rapid bleeding such that the central nervous system is denied blood and functions essential for life are shut down.
The quickest way to achieve neural disruption is a brain shot BUT the brain in a deer (or coyote for that matter) is not a large target and is highly mobile. A misjudgement in aim or if the deer moves its head as the shot is taken or while the bullet is in flight and you will most probably inflict a horribly mutilating injury which will allow the animal to escape and will cause a long slow death for the animal. Most people aim in the wrong place for a brain shot with deer ... the aim point is ABOVE the eyes and should only be attempted at short range by a rifleman who is very confident about his ability to place his bullet accurately in the correct location.
A neck shot will allow for a greater range in bullet trajetory and is therefore better as a longer range target. Shock from the bullet will be transferred to the central nervous system through bone and cerebral spinal fluid and will result in instant coma, during which essential functions such as breathing and heart regulation will cease ...... the deer will never recover its conciousness (i.e Dead Right There).
The other target area is the front area of the lungs. This portion of the lungs is richly supplied with blood. If a bullet passes through the front portion of the lungs it not only destroys lung tissue essential for oxygenating blood, it ruptures arteries which, together with the animals efforts to breathe, fill the lungs with blood. The animal both suffocates because the blood stops air from getting into the lungs and bleeds out because the blood is being pumped by the heart into the lungs. Blood pressure rapidly drops, blood is no longer supplied to the brain, the animal looses conciousness and dies rapidly (within 30 yards of where it was hit).
Do not make the mistake of thinking a heart shot will kill an animal quickly. If you make a perfect heart shot and destroy the heart completely, there will be some lung damage but minimal bleeding into the lungs. The brain will still have a supply of oxygen, the muscles will still have a supply of energy, the animal will bolt without leaving a blood trail (no blood pumping remember) and it will be able to cover upwards of 100 yards or more before the central nervous system fails. 100 yards in heavy cover with no blood trail can mean a lost animal ...... dead, but the hunter may never be able to find it.
The hilar zone (the front part of the lungs, the major arteries feeding the lungs and a plexus of nerves supplying the lungs and heart) is in the lower front chest area protected to a large degree by the shoulder joint. A shot into this area can achieve disruption of the locomotor system (break one or both shoulders), rapid bleeding assisted by blood pressure and breathing action, suffocation by blood displacing the air in the lungs and preventing any air from entering the lungs, and some neural disruption to the vital cardio-pulmonary system. This is a very safe target zone which will always result in a rapid death.
A shot which ruptures an artery such as the femoral will result in death, BUT, arteries are flexible and resistant, to a degree, to pressure (they have to be to handle blood under pressure form the heart). A direct or nearly direct hit on the artery is essential to guarantee arterial bleeding. The artery is a small target with nothing to distinguish its location externally. In the time it takes the animal to bleed out via a femoral arterial bleed it may well have travelled a considerable distance. I refer you to a book by Matt and Bruce Grant* where Matt Grant relays:
I once wounded a deer in the hind leg at considerable range and lost the trail after a mile of diligent tracking. Two days later I found him dead 3 miles from where he had been wounded. I found that the bullet had cut the femoral artery, the large artery on the inside of the thigh. This is possibly the fastest bleeding muscular artery, yet that deer had travelled 3 miles.
The best target will vary on the range to the animal and its presentation (standing broadside, angling toward or away from you, head up, head down, etc) and the abilities of the person making the shot. I would always advise attempting a hilar zone shot as a first priority target. There is reasonable margin for error and will result in a rapid death. You should be aiming for the off side shoulder about one third of the way up the chest. If the deer is angling toward you you'll be aiming roughly where the lower neck joins the chest (and projecting through to the offside shoulder), if the animal is angling away from you you'll be aiming somewhere just behind the nearest front leg (and projecting through to the offside shoulder). At extreme angles away from you, you may end up aiming at the rear ribcage (and projecting through to the offside shoulder .... all at around one third of the way up the chest.
If you are more confident about your field accuracy (your ability to place a shot under field conditions on a living breathing target can be very different to your ability to group holes in paper under perfect conditions) and your knowledge of the animal's anatomy then a neck or spine shot might be preferably. If you're very confident, you know exactly where to aim and the range is short (within 25 yards) you might try a brain shot.
Spinner
* "The Sharp Shooter: How to get the best out of rifles and ammunition", Matt & Bruce Grant, AH & AW Reed (publ), 1972