Never get a puppy because a child begs for one, no matter how mature or reasoned her arguments. The idea of buying a living creature as a Christmas present is really bad, although I see you're not actually planning to get one as a Christmas present. It's not like buying a mobile or the latest new craze. It's not a commodity it's a live animal with needs and fears.
Also as pp said you cannot time precisely when you would get the puppy if you're getting one from a decent breeder and six weeks is not time enough to settle a puppy into a home before planning to leave it alone; even if you get an 8 week old puppy at the beginning of the school holidays it'll still only be around 4 months old when you're planning to leave what is a very young baby animal alone for hours on end. That wouldn't work and if you did it would be the height of cruelty, very possibly resulting in a dog with major separation anxiety and other emotional problems.
It doesn't sound as if your home setup is at all suitable for a small puppy to be brought into.
An older rescue dog might be a better bet and you might just get away with a six week settling in period if the dog is coming from a decent background.
What happens though if she actually decides that she's rather bored by the dog after a few months? Dog's need training, they need walking when it's dark out on a cold morning before school, every single day regardless of the weather, and again in the evening, when perhaps she'd rather be out hanging around with her friends.
Dog's are wonderful animals but they are a huge commitment for many years.
Fourteen year olds can be very fickle. Are you prepared as the adults to take care of the dog for the rest of it's life. All the expense, vet's fee, insurance, caring for the animal if it is unwell, all the walking. Making sure it's not left alone too long. Not being able to just go out for the day or going on holiday without thinking what you're going to do about the dog.
You can't expect a child to think of these things, but as adults you should and ultimately it all falls on the adults in the family, especially if a child loses interest.
And if she does lose interest, she wouldn't be the first child who has begged for a pet, sworn she knows what's involved in looking after it and promised she wants and is prepared do it all, and then lost interest after a few weeks.