I'm a Vietnamese citizen and currently living in Japan. Can I apply for a South Korea tourist visa at the Korean embassy in Japan, or must I do it at the Korean embassy back in Vietnam?
You should be able to apply from Japan
If you are currently a resident of Japan, regardless of your nationality, you should be able to apply for a visa for the Republic of Korea (henceforth South Korea). Some countries do have the restriction of having to apply for a visa from your country of citizenship (henceforth C.O.C.) rather than residence, however South Korea does not apparently follow this procedure.
There is however, a chance that your application may bear higher fees (and in rare cases lower fees) and a longer processing time if you choose not to apply from your C.O.C..
Looking at the South Korean Ministry of Affairs website for visas, it appears that a Vietnam citizen on a normal passport is exempt from visa-free travel, however it does not appear to state that applying from a different country will garner any additional fees or wait time that applying from Vietnam.
Depending on your length of stay the prices for the visa is...
Less than 90 days US$40
More than 90 days US$60
Double Entry US$70
Multiple Entry US$90
As is always then case with visas, call or email a consulate of the country involved and confirm these details for your individual case.
A list of South Korean Embassies with phone numbers and email addresses, can be found here.