omfg why all the snark?!
OP your post was as reasonable and non-inflammatory as they come, I cannot explain what has happened in this thread but I apologize on behalf of everyone who's been so unnecessarily rude.
No driving is understandable - they probably don't screen their on-call nannies for insurance/clean driving records/etc. as that's a bigger cost to them, so they don't want the liability of having them drive on the job. The walking is weird and inexplicable and I've never heard of that. As PP said, with a toddler 15 minutes of walking might not even get you out the driveway!
My agency DOES allow on-call nannies to drive, but we go through the same screening process as FT and PT nannies do, so it might just be that agency?