There are no required vaccinations for those visiting or moving to South Korea. There are also no major health risks that you should be worried about – at least, no health risks that wouldn’t be found in any Western country.
It should be noted that Seoul’s air quality is extremely poor, especially in the early spring when the toxic yellow dust from China comes to Korea. It is very common to see locals walking outside with masks to cover their nose and mouth, and many people will tell you that it is not safe to run outside. If you have breathing problems or problems with asthma then you may want to ask your doctor about any special medications that you may need prior to arriving in Seoul. Alternatively, you could consult a doctor here as they are extremely well equipped and quite used to dealing with this problem.
When the yellow dust comes you will want to be prepared for it. Visit this website for more information about the yellow dust: http://en.wikipedia.org/wiki/Asian_Dust