Does anyone know how to get a copy of an award that's not on your OMPF?

The orders for your award should have been uploaded into iPERMS by the unit that gave it to you. That would be the first place I would look. https://iperms.hrc.army.mil/rms/. Good luck.

SSG (Join to see) do you have any paperwork for it at all? A DA638?

The issuing S-1/G-1 should have the Order Number in a new system called TACCS. See excerpt from AR 600-8-105 below. (However, this may not have been in effect when you were a private)

b. Commanders of Active Army detachments, companies, batteries, or battalions that are serviced by a centralized MILPER office will not issue written orders. When written orders are required,
commanders who are not authorized to issue written orders will submit a request for orders to the commander responsible for providing administrative support . However, the installation adjutant general may authorize those battalion and brigade commanders with automation capabilities (such as the Tactical Army Combat Service Support Computer System (TACCS)) to publish orders for the actions listed in (1) through (4) below.
(1) Awards.
(a) Good Conduct Medal.
(b) Diver Badge.
(c) Drivers Badge.
(d) AAM per AR 672–5–1, table 2–3.
(e) ARCOM per AR 672–5–1, table 2–3.
(2) Intrastation reassignment (no travel).
(3) Attachment or relief from attachment (no travel).
(4) Special pay (flight pay and sea pay).

Did they unit not have to send it up to higher? They can keep records longer- maybe electronically.

I recently had the same issue for an award I received on active duty that was never uploaded. I contacted HRC and they assured me that if they award showed on my 214 my commander could regenerate the award.

Try this website. This is where all documents are held like your DD214, awards and other things electronically.
http://www.archives.gov/veterans/

You have to know the preset it was issued under.

I had the same problem with some awards, such as the humanitarian medal. What HRC told me is to have someone high up write "something official" (their words) to put in my iPerms. I wrote a memo and I contacted my former 1SG, who is now a CSM, to sign it. It worked. Also, if it's on an NCOER, that works too. That's what they told me.

Normally when an award is presented, the green folder has the award certificate in one side and the 638 on the other. It appears this is not the case for you. Contact the unit S1. Unfortunately, if there is no source document, then there is no award.

If all else fails, make an attempt to find the person that recommended you for the award, the company commander, and or the the approving authority.

Good luck.

Also, if the award was recent, there should be a record of it in HRPAS as well as iPERMS