The minivan taxi services that serve stores like Megamart and some Homeplus should be able to help you. I've used them twice to move a sofa and bunch of boxes.
English depends on the driver you get.
I have a central number which is 051-644-1717 and the card I have has the driver's cell phone 011-876-9983. Alternatively you could just turn up to Megamart in Namcheon and chat to them. Face to face is always easier I find.
My Korean ain't great, but I think could you move a fridge w할ould be somewhere along the lines of 냉장고 이사할수있어요? not sure if that's the correct can / be possible for this context though.
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