When you are at Naejangsan National Park, there are several food and drink stores located near to the start of the major trekking routes. So when you are there, you can buy some food and cold drinks for your trek. However, the price here will be more expensive so an alternative is for you to buy them while you are at the Naejangsan tourist village.
There is also a food and drink store next to the cablecar station near the top of Yeonjabong Peak. This store comes in very handy for people trekking in the mountains because it provides a good resting stop, with food and cold drinks (especially energy drinks such as Gatorade).
There is a Family Mart (24 hours convenient store) located just next to the bus terminal at Naejangsan tourist village. This Family Mart is an important place to buy various foodstuff e.g. biscuits, bread etc as well as drinks such as fruit juices, milk etc which will help especially for those on a budget. However, be take note the things in South Korea are generally not cheap and can be comparable to prices in Japan, Europe or the US.
Attached are more photos of my meals at the Prince Motel at Naejangsan National Park. In order to include the meals in the stay at Prince Motel (which is 30,000 Korean won per night), I have to top up about 15,000 Korean won per day, which I think is worth it considering it is for breakfast and dinner (lunch is taken at the national park).
When I was at Naejangsan National Park, I had most of my meals at the restaurant in Prince Motel (this is the motel where I stay, please see tips at accomodation section). The meals are local food, consisting of many side dishes including Kimchi. The main dishes would be rice, miso soup, eggs and a meal dish (usually fish or pork). There is also ice water being served.
Overall, I enjoyed the food because it is very delicious especially the sweet minced pork cooked on a stone pot which is tasty. Also, the meals are served hot, which is very good especially when the weather is cold outside. The restaurant also has soft drinks such as Coke. Please see part 2 of this tip.
Being a major attraction in South Korea, there are lots of local restaurants at the Naejangsan tourist village area. They are lined up in rows along the main road and you cannot miss them. All of them serve traditional Korean dishes which normally include Kimchi and lots of side dishes, and the main dish which usually consist of rice, soup and meat (e.g. fish, pork). Price would be around 10,000 to 20,000 Korean won per meal, depending on what you order.
The main problem would be the language barrier as most locals cannot communicate in English, however there are photos and prices of the various dishes on the menu so you just need to point to them what you want. The locals are very friendly and laid-back so it should be OK :)