I had a similar problem a while ago that may come up again.
My ILs suggested taking DS out in the car with them. Although I think their car is no safer than a carboard box TBH. They did take him once when he was a baby and when they brought him back, the straps where done up wrong. When I pointed this out, MIL said "oh well we can't be expected to remember everything". I was fuming at her attitude (my DS was weeks old at the time). After that I dropped him off and picked him up, although they haven't had him for nearly a year now.
They are getting a new car and made a comment about a second hand carseat, which we said no to. Then they told us their new car has some sort of special fitting for car seats. But my FIL has parkinsons and I am not comfortable with having DS in the car with them. I know the new car is for power steering as he is having trouble with the steering on his current car, but even with the power steering, I have seen how tired he gets and they were saying about how he struggles to do his zips and stuff and I see how much he shakes and I am really not happy about DS going with them. If the subject does come up again, I am going to get DH (who agrees with me) to say something, not sure what yet, but it is not going to happen.
YANBU at all. Their age would make me very wary and my ILs are not as old as yours but I know they would have trouble keeping up with DS. MIL comments on it from sitting in her frontroom so I definately don't trust them to take him out yet. I don't feel they take safety etc seriously. If we go over to her house she never shuts the kitchen door and everytime I do, next time she goes into the kitchen, she just leaves it open again while she is sat in the front room. Obviously their kitchen is not at all child proofed and I don't feel she takes it seriously.
When it comes to your child, always go with your instinct and I don't give a damn if I upset anyone when it comes to the safety of DS.