I've had several experiences with car problems whilst in KNP, it's best to be sure that you have local — Nelspruit or Phalaborwa — phone numbers for whatever car hire company you use.
Just the reassure you, I'll detail my experiences:
1. Years ago (I think it was 2000), I had a flat near Olifants (where I was staying) and rather than waiting for another passing car, I drove the 8-9Kms back to camp, which destroyed the tyre.
The petrol station attendant at Olifants removed the tyre and installed the spare, but as I had another few weeks in the park, I needed to replace the ruined tyre.
This was before everyone had cellphones, but I was able to reach Avis at the Phalaborwa airport (using the pay phone and a local calling card!) and drove there next day.
They replaced the spare (and charged me for the destroyed tyre), but also discovered that there was a fault (warning light) with the car I had, so they replaced that as well.
2. In 2006 I had several car problems.
I was renting through the excellent consolidator AutoEurope (which you might check as their rates are very good) and my car was actually from National/Alamo.
In the far north (Parfuri) one of the car's electric windows jammed and couldn't fully close — which can be dangerous, particularly when you're around baboons.
I contacted National/Alamo (at, I think, their Polokwane office, which was the closest) and they arranged to meet me at Mopani (my next camp) with a replacement car.
Later in that same trip, when I was staying at Lower Sabie, my car died totally near Skukuza.
I was out of emergency cellphone range, but fortunately a passing motorist reported my situation and help arrived and I was towed back to Skukuza.
A replacement car was sent from Nelspruit, but only arrived after gate closing — we had arranged the late entry for the car-hire company — so I had a long, long wait at Skukuza that day
. I then had to pay a fee to be escorted back to Lower Sabie after gate closing time.
The plus on this "adventure" was that the second leg of my "escort" (from Nkulu to LS) was provided by a ranger in an OSV and en route we had a superb sighting of a female leopard!
Next day I discovered that a major warning light was flashing in this replacement car, so I rang National/Alamo at Nelspruit yet again and several hours later they met me at Crocodile Bridge (seemed to me better to take a drive to Croc Bridge than to wait in camp at LS) with a third replacement car.
And on my return drive to LS, I had only my second ever cheetah sighting in KNP — which I likely would not have had if I hadn't needed to go to Croc Bridge to change cars.
Since then (KNP visits in 2007 and 2008), I've not had any further car problems.
But I don't think that you need to worry unduly.
As long as you have contact numbers for you car-hire company (and I expect their national emergency number would connect you with the local office as well), you'll be fine.
(And if I can help with any other advice for traveling from the US to SA, please ask!