Better yet, why don't you potty train your kids before they are like, I don't know, 4 years old?
But I know, when parents use Pampers, the kids like them so much they don't want to go potty when they are supposed to.
A wonderful snarky comment, but all kids potty train differently.
That being said, mine was done by 3, though she still had "emergencies" until she was 5 or so.
Anyhow, the OP never clarified the ages of the children, merely asked a question about whether it's safe to leave the bag on the stroller rather than carry it through experiences, and I'd say, overall, it is. Just don't make it a nice one, and don't keep it overly visible (put it in the stroller seat while in lines), and don't keep any irreplacables / special items in it...just diapers / wipes etc...
Also, please don't leave a (even clean) diaper bundled up on it. That's a fun joke to josh about, but in reality is rather disgusting to see.
If she's really that concerned, they do rent lockers near the front of the park, and that's a perfect solution.
Also, unless your kid is a pooping machine, you really have no reason to carry a whole lot of supplies. When my kiddo was in that stage, I could get by with a very small bag that had 2 diapers, a thin ziploc with a few dozen wipes, and a few other items. Granted, I never used that setup at Disney (didn't take my kiddo until she was potty trained), but I did use it at various zoos and museums and other things.
There's no need for the massive army pack diaper bags that I've seen unless your kiddo processes food extremely quickly, or eats taco bell as a baby.