You will need:
A bed
A lead - please, please, please don't get an extending one, they are a danger to the dogs they are attached to and other walks. I hate them with a passion. It is almost always dogs on extendable leads who bother my whippy when we are training her to be confident around other dogs after she was attacked by a dog on an extending lead. The dog ran at her with such force it broke the lead and the owner was left with no way to reel his dog in. They are, imo, a flashing neon sign that says selfish and clueless dog owner (apologies to anyone who uses one, but that is the experience I have had of them in my area)
I rate Halti training leads for regular walking, if you want to give her extra freedom once you are in open spaces and to train recall you can buy extra long training lines, but be aware that for safety these should be attached to a harness not a collar and not one of those anti pull harnesses.
Wrt the pulling a harness is not really a replacement for training - The Complete Idiots Guide To Positive Training lists some ways to train loose lead walking using a clicker and there are lots of YouTube clips that show you how you can do it, of course halti's help but should be used in conjunction with training - not instead of. I prefer the halti harness to the head collar, most dogs seem to take to it better than the head collar.
At least two dog bowls.
A supply of the food the dog is currently - you can change the diet but do it gradually mixing in a handful of your chosen food and increasing that amount slowly.
A crate if you are using one - If the dog is not crate trained already you will need to train it to accept the crate before locking it in there for long periods fo time.
Insurance
Your local vet's number
Details of your nearest positive training class
Kongs, Nyla Bones, other toys the dog might like (ask the fosterer)
Books - The Complete Idiots Guide to Positive Dog Training is really good and covers everything, but there are loads of others recommended on here all the time, I also found Karen Pryor's book on clicker training helpful.
A clicker and treats if you decide to go down that route of training (I am in love with clicker training atm, it is like magic)
I wouldn't put a new dog in the back seat with children to be honest, my Dad has dogs in the boot with no grill, he just raises the head rests up as high as they will go on the back seat and that keeps his dogs where they should be, if not then go for the crate option.
You shouldn't really need to stop if the dog is adult but if you want to stop half way through the trip then the dog will appreciate it.
I am sure there will be things I have forgotten.