Caroline Cullen is a doctor because she wants to be a doctor and it's what she's good at and whatnot, but having access to a human blood supply is one of the perks. However, this can only come into play when there is an excess of blood (sometimes there is) at such points she can fudge some numbers and get more blood than needed and have it disappear unnoticed.
The majority of the time, though, there isn't a definite excess of blood, and thus the blood she could get would come at the expense of risking people who needed it not getting it. Thus the Cullens don't drink human blood most of the time. In theory they probably could without killing anyone and without attracting attention but there are morality issues of bodily autonomy involved.*
So most of the time they hunt. Not as Edward hunts because that sort of bloodsport is pointless and unsustainable. Apex predators are few in number, eating them on a regular basis is going to deplete your food supply and fuck up the entire ecosystem.
If apex predators are the most healthy for Twilightpires (did not know that until this thread) then the invention of the tranquilizer dart must have been a cause for great celebration (right up there with the invention of colored contact lenses) for Caroline's entire clan.
Knock out the predator, siphon off some blood, pass any useful information gathered by examination of the animal on to friends studying such (they've been to University enough times that they must have friends in the field) who have agreed not to ask how the information was gathered (on the understanding that no animals will be harmed beyond tranquilizing) for the betterment of science. Drink the blood at a different location to avert the off chance base instincts will take over and cause a feeding frenzy (never happened, never seemed likely to happen, but safety first.)
Before the ability to tranquilize made sustainable feeding on apex predators a possibility the Edith-verse's Cullens probably sustained themselves largely through making sure to be within walking distance of a slaughterhouse. Sure, farm animals might not be as healthy for vampires as wild caught apex predator, but it's not like someone else was doing something useful with all that blood.
In Twilight, and many other things as well, vampires can't eat solid food. They're on an all blood diet. This presents an interesting situation: they must feed on animals (humans are a type of animal, remember); there is absolutely no reason they ever need to kill one to feed on it.
Vampires could theoretically be on the equivalent of an all fruit diet: things may be inconvenienced (in theory the covenant with a fruit producing plant is that you eat the fruit and then your droppings distribute the seeds, anyone who uses a toilet is breaking that) but nothing needs to die or be dismembered.
Plus: poetry. Edith's family would very much be on Mr. Squid's side, and having a way to escape Mr. Sperm's reality (There's only one reality: / All creatures who'd succeed at sea / Must murder for their daily meal. / What else can all your thoughts reveal?) would be welcome to them.
Feeding is a morally fraught topic. But for vampires it shouldn't be. Blood is not flesh. The difference between the two changes everything. If meat identical to that produced by killing animals could be created by not-killing/maiming animals, the only reason to kill animals for meat would be if you wanted to kill things. That would be a completely different world.
Whether the animal is alive or dead at the end of feeding doesn't change the fact that the blood you're getting out of it is the same blood. We already accept the tranquilizing of wild animals for various reasons. If the blood is removed by medical-based means rather than fangs you eliminate both the possibility of vampire lions and tigers and bears and the possibility of becoming a disease vector (vampires with sterile needles: now 100% less likely to transmit Lyme disease)
* Vampire bats often feed on their victims without the victims ever realizing they've been victims, there's no reason that style couldn't be adopted by actual vampires. If Edward can nightly break into Bella's room without ever being noticed then Twilight-vampires could probably show up in the rooms of sleeping humans with topical anesthetics, syringes, and a knowledge of where to draw blood such that the puncture wound won't be noticed.
It's just that, even though that leaves the victims alive and unaware, that's really not what moral people would do. That's more the stuff of alien abduction stories than good guy stories.