The text does not give a direct answer to this question. That is not to say we must be left wondering! There are two common reasons why a text may omit details that would seem relevant to us.
- They are simply not relevant to what the author was trying to communicate.1
- It is assumed that the immediate audience would have understood the scenario given their knowledge of the contemporary context.
I would suggest that this is a case of the latter. It fact I would go farther and suggest that knowledge of the contemparary scenery is unnecessary. We don't need to know the exact lay of the land or details on wells and other water sources in order to understand what transpires in this passage. Applying a little bit of pragmatic awareness to the situation will leave us with no question at all.
Drought is not the complete absence of water.
If it was, everybody would be dead. Mammals cannot survive for more than very brief periods without water. Rather, just as famine can happen even when there is some food but just not enough to go around, so a drought can mean simply a water shortage. There isn't enough to meet all the needs.
The passage in question clearly indicates that God was in charge of the local weather patterns and that the state of 'drought' was a direct result in it not raining when in otherwise might have. This is actually a common problem even today.
I just got back from an area of sub-Saharan Africa where subsidence farming is still the normal way of life. One region I visited was stuggling with a season of drought. It is normal for it not to rain for a time, but last year's rainy season came early and the local water table did not get replenished when it needed to be. As a result the region's water table has dropped. The local villages are surviving because they can carry water from the few wells in the area that are deep enough to still have some water, but the extra effort, distance and time mean that—while they can still drink every day—they are unable to sufficiently irrigate their crops. They are still raising some produce in small gardens close to water sources, but they are also eating through stored crops to survive the year. If such conditions last, it will turn into famine. Caused by drought. The is water in the area, just not enough.
If you consider a scenario like this one, the story you reference makes perfect sense. This is not a definative answer on where the widow got her water. Rather I would only conclude that this question does not raise any real difficulty in the text. The lack of rain and state of drought would quite naturally make food scarce even while there was some water available. Water was undoubtedly precious, but obviously the situation was not so acute that nobody was drinking (or there would be no-one to tell this story) however the amounts needed for drinking are a drop in the bucket compared to what is needed to effectively raise crops.
That the widow could come up with a jar of drinking water raises no concern about the text or the state of drought in the land.
1 In Christian theological frameworks this concept is generally extended to God (who is believed to be the ultimate author of the text). If the human authors left out a detail is often taken to mean that the God does not consider it necessary or important to know that detail to understand the point of the story. This answer will do nothing to explain the point of the story, only note that it is plausible.