Is there enough time to discover every location and do every side quest before finding and handing over the water chip?

Finding the water chip is only one part of the game, there's another side to the main quest after that, where you don't have a time limit.

The two playthroughs that I did I was able to explore every location without sweating the time limit much. A good idea would be to recruit Ian over at Shady Sands, the first town you should hopefully be running into. He can give you a good idea of where things are, which should cut down on any aimless wandering that could waste a fair bit of time.

I would say there is enough time, but as has been stated the game isn't quite over when you hand in the water chip. After you hand it in, you'll have plenty of time to explore

Sidenote; When exploring always be sure to carry rope, rad-x and radaway to avoid any inconvenient radiation related deaths.

well i was in vault 15 already, -no waterchip.

i think im gonna head to the Hub or to Junk Town now since the only quest i found in Shady Sands is the one with the rad scorpions and the one with the raiders but they are currently too strong for me

Of course not, do you think they would put the reason for the game's MQ in the 3rd place you visit?

people in 1998 thought differently