It's without the -ing if it's an order to do something.
It's with the -ing if they're deciding to do it
Essentially it's the difference between whether first noun is passive and the second is active, in which case it's ing, or whether the first noun is active and the second is passive, in which case it's not ing (using first noun and second noun because I don't know grammatical terms)
"I had John fix the car" means you had John fix the car through your direction.
"I had John fixing the car" means the fact of your presence had John freely choosing to fix the car. "I had John fixing the car in no time" means your presence enabled him fixing the car to be the case, whereas "I had John fix the car in no time" means you ordered him to fix the car quickly rather than slowly. (Technically, because the idiom doesn't transfer, so quickly it defies the laws of physics.)
"The ample supplies had the government urging vaccinations" means the ample supplies had the government choosing independently to urge vaccinations.
"The ample supplies had the government urge vaccinations" means, the ample supplies had the government urge vaccinations rather than look for more vaccinations because, as the directors of societal welfare, they decided there were enough vaccinations to go around.