No, you probably won't have much problem cleaning the carpet.
You see, "water stains" are most commonly caused by a water leak that drips through a drywall or plaster ceiling, or maybe down a drywall or plaster wall. Gypsum and/or calcium carbonate from the drywall or plaster (respectively) will dissolve in that leaking water. Then, when the leak is fixed and the water evaporates out of the carpet, that gypsum or chaulk that was dissolved in the water will be left behind in the carpet nap. That's why they tell you to use vinegar to clean it up with; cuz it's a mild acid and will dissolve the gypsum or chaulk that remains behind.
In this case, however, what I'm seeing is that the water is coming UP THROUGH the backing of the carpet, not dripping onto it from above. You might get some whitish staining of the carpet just cuz of the stuff dissolved in the water, but any dirt in the water (as you might expect if there were a roof leak) would be filtered out by the carpet backing, and so the most you might get inside that carpet would be those gypsum or chaulk deposits, and they're relatively easy to remove with a weak acid.
So, it shouldn't be hard to get the carpet clean once it's dry.
But, I'd say to contact your neighbor and see if this has happened in other years. In the mean time, use a wet/dry vaccuum cleaner to remove as much of that water as you can.
PS: Where I live, they'd never put a water supply or drain pipe in an exterior wall because of the concern that it could freeze (and burst or clog with ice, and also burst). Would they do that in Tennessee? When the guy left the house, did he shut off the water to the house and drain the lines?