Universal rents that DeLorean from a private owner. There are many replica DeLorean Time Machines across America, most built by fans; so it was easier to just rent one, rather than buy another one that the studio would have to pay to maintain themselves.
The one located next to the train is in dire need of restoration. The team that restored the original "A" car, which is now located at the Petersen Automotive Museum in Los Angeles (
DeLorean DMC-12 | 1981 DeLorean DMC-12 Time Machine), requested to restore the DeLorean in Orlando also, but the park for whatever reason declined.
Just FYI, there's also a DeLorean Motor Company located in Orlando.
DeLorean Motor Company Florida | The Best Source for your DeLorean in Florida which could easily help with the restoration of the car.
On another side note, Universal destroyed the original tender behind the locomotive back in the early 2000's, just to make way for the new Men In Black ride, rather than move the entire train itself; easier to just destroy it.
And that was an original screen-used movie prop vehicle by the way, that if sold on the private market, would probably been in the high five, or low six figures.