What do i do

It just could be that she's using the car seat issue instead of saying directly that she doesn't want her mother (or perhaps anyone) driving her children anywhere. Any car journey is a risk and some of us, especially as we age, are not particularly good drivers. I hated the way my father used to drive and would do anything rather than let my kids go in the car with him.

Maybe she wants to give up work?
Are they relying on her income to keep their heads above water?

Maybe she just feels sad about something and needs a little moan to someone who loves her.
One of my daughters likes to vent a bit by sharing her problems. I have finally discovered that what she wants (usually) is NOT for me to solve them but to acknowledge her distress.
I now say, pretty straightforwardly, "do you want me to think of solutions or do you just want me to listen".

That often generates laughter and "I just want to moan".
I really love her. I think we are very close - but I'm known to be a "problem solver" - and not everyone wants or is able to fix the real issues.

Our older grandchildren are all teenagers now but our youngest son is quite a lot younger than his 2 siblings so unsurprisingly we now have a delightful 2 year old granddaughter. Our son works with us in our small family business in our home and his partner is a nurse working 3 long (12 hour) days.
They do have a childminder but she doesn’t work every day so if working days clash he brings her here so the 3 of us try to entertain her and keep business going between us. There are other times when we’ve had her for a morning or afternoon on our own but they usually bring her here. We have now bought a car seat but we get him to fit it as it seems a struggle but we’ve not used it very much, I should add they live nearby. At the moment she spends a lot of time in the garden when she’s here but I know as winter draws in she will get bored stuck in the house.
Part of the issue is that now DH and I are in our mid 70s and I have health issues and struggle to keep up with her energy levels and worry that if I took her out on my own and she decided to do a runner I couldn’t keep up with her. I’ve also lost a lot of confidence with driving so not sure I’d feel comfortable driving her anywhere - not sure her parents would either to be honest.
Unlike Horsewrench they aren’t saying they have any issues and her maternal grandparents are much younger than us so they do a lot of picking up from childminder, taking to parks etc. Do the other grandparents help with your family Horsewrench?

I really like NotSpaghetti's comment:
"I have finally discovered that what she wants (usually) is NOT for me to solve them but to acknowledge her distress.
I now say, pretty straightforwardly, "do you want me to think of solutions or do you just want me to listen".
That often generates laughter and "I just want to moan".

I am going to steal this idea!! I, too, am a known problem-solver. And I need to reign that in. Thank you!