First, you must realize that you have to have the proper visa. If you are a full time student and have the appropriate visa to do so, you will have the right to work part time in S. Korea. After that, resumes are very important to Korean. They don’t look like a resume from the west. They actually look like a form that you fill out. They have, basically, the same kind of information, but it is all on a kind of chart and very abbreviated. Your resume needs to be in Korean. and be only one page. The main difference besides the look of the resume in the west and in Korea is that in Korea, they want a picture of you. I am an American, and if I put a picture of myself on my resume in America, they will reject it, but in S. Korea, they want a picture of you in the upper right hand corner. A passport type photo will do, but a snap shot can do the job too. What you look like when you are hired is very important in S Korea.
I have never applied for a job at a convenience store in S. Korea, only at schools. However, as a university professor for several years in S. Korea, students have shown me their resumes. Letters of reference will also help. If you have a letter of reference from your last job, I would call it mandatory. They take work history seriously. Often, they will ask of a letter of release from your last boss before they will give you a new job. If you have letters of reference that are character references as well as work references, it will be important to S. Koreans. Remember that S. Koreans like Christianity, so if you have a good relationship with a preacher, that kind of letter will be very welcome.
Since I have never applied for a job at a convenience store, if I wanted to do it, I would go to the store where I wanted to work and ask some questions. The store could belong to the person working there or someone else, and they could tell you who you should apply to in order to get the job, When my daughter became an English tutor, she actually had to go to the neighborhood center and receive some sort of teaching certificate. You need to check with the neighborhood government center, the jumin center, and see if there is any paper work you need to do there to work in a store as a foreigner. They like to keep track of everything going on in the neighborhood.
The only place I have actually seen a foreign student work in a store is in Itaewon. At the import shop, there were several foreign students. I would probably go to one of the import shops in Itaewon, maybe the one next to The Foreign Food Restaurant and ask some questions. Those students know how to get jobs in S. Korea, and they are all foreigners. I would be asking a lot of questions and learning where to go, what to do, and who to talk to. I can only tell you a small amount from what I learned from my students at the university, but those guys could tell you much more than I can tell you. If you go to the main subway in Itaewon, stay on the side away from the big hotel across the street, and go the direction of the park, there is a Turkish food place on the corner at the next block. Turn right there, and if you keep going, there will be an import shop on the right and a little further down on the left. The one on the left is larger, and that is where I knew foreign students working at a store. They can give you much more information than I can give you. I hope you get your job. There are also other kinds of jobs foreign students do in S. Korea. I have seen them work at restaurants, work in factories, and I have even known about artistic students who worked drawing pictures in S. Korea. There are also private schools in S. Korea who look for someone to work part time teaching English, but you need to be sure to be a native speaker of English to apply at one of those. One of my Bangladeshi students got a job working at one of those, but had to quit because they could never justify the visa because Bangladeshi is not a country where there are native speakers of English.