Do you need orders in order to wear the OSR or do you just need you deployment orders?

You need proof of a creditable overseas assignment. So yes, in this case you'd need a copy of your deployment and redeployment orders.

Wouldn't hurt to provide them a copy of your end of tour award as well.

Has anyone asked the OP what the dates of his Korea assignments were? They no longer award the OSR for being stationed in Korea, but instead have the Korean Defense Service Medal.