After doing some research, it appears that you actually have to have your documents legalized by an apostille certified by your government (US in this case).
From esljobproject.com (emphasis mine):
If you are Canadian, criminal background check record Apostilled by the Korean Consulate is required. All other nationalities require a criminal background check Apostilled by your government.
An apostille is a seal applied by the Department of State in the U.S. to authenticate a document in a foreign country in order to assess the authenticity of an official signature. The Apostille is recognized as form of an international notary seal. An Apostille can be used if both countries (the country issuing the document and the country in which the document will be used) are part of the international "The Hague Apostille Convention".
The only exception to this rule is Canadians whose government does not certify apostilles; Canadians have to have their documents notarized at the Korean embassy.
A Google search for "apostille us" turned up a number of services in the US that will apostille documents and return them via post, though I am not familiar enough with any to make a specific recommendation.
You might be able to get your documents apostilled at a government agency (though some/all of them might require that you show up in person);