Unfortunately I think all too often these days, parents and grandparents pay out for too much for the kids, meaning that they fail to grasp the concept that you CAN'T have everything, and that if you want something, you need to work for it. Hence we now have a nation of entitled young people, who think the world owes them a living. Yes, I know it's nice to treat our kids and grandkids, but personally, unless asked for a loan, which if we agree, is set up with a legally binding agreement, and paid off by Standing Order every month, we tend to only pay for things like meals out, drinks, etc. we would pay for other small things, but due to disability can't get out much, so it's not something which arises. Youngsters need to realise that there isn't always going to be someone there to bail them out, so they have to learn to stand on their own two feet.
So, having said all that, it will come as no surprise to the OP, that I totally agree with her DH. Let your granddaughter work, and earn the money to buy her own car. If your DH agrees, you could perhaps offer her an interest free loan, but if you do, write out a contract, noting the full amount lent, and the amount due each month, and get her to sign it, as she would have to do if borrowing from a bank or other institution. She then needs to set up a Standing Order, and tell her that failure to pay, will mean you take her to the small claims court, and MEAN IT! (or at least make it look like you do, so that she takes the whole thing seriously) If you can't do this yourself because you're used to spoiling your grandkids, then get your DH to do it, as he'll likely come across as meaning what he says, whereas she may think you'll be a push over.
If she doesn't want to do this, then she needs to get a job where she can use public transport, as we all did when we were young.