Your question is “do you need to?”, need being the operative word. As Azygous kind of pointed out, it depends.
If it is dark in the coop at night the chickens will not eat or drink. If they have enough light they probably will. That light could be from something you put in there or from an outside light, such as a street lamp or security light. They don’t need to eat at night, it’s best if they are in the dark from a few different perspectives. They need their dark down time but some get light anyway and do OK.
When the chickens wake up they will be hungry and thirsty. If you leave them locked up in the coop for a fair period after they wake up, the food and water should be in the coop. If they can get to it in the run, it can be in the run. I don’t have a magic number for how long after sunrise you need to let them out to get to food and water in the run, I’ll just say the sooner the better.
Some people feed and/or water in the coop. Some feed and/or water in the run. I do both. We all have our own reasons. I almost always have younger chickens in the flock so I set up multiple feeding and watering stations, in the coop and outside. With food and water in the coop, if I want to sleep in a bit instead of getting down there early to open their door I can.
There are many ways to do this, decide what works best in your situation.