Does anyone know how to submit an award (Navy Achievement Medal) for a service member that has been medically retired for over 10 years?

GySgt Bill Smith - Just some thoughts:

1.) I'm assuming by your profile that you're still active duty. It might be worth initiating contact with the Marine's former command. BnCO's and Regimental CO's can do "pocket" promotions & awards. Minimally, they could verify their dates in theater and the Marines connection to their unit.
2.) Also, if you are aware of the Marine's current residence, you might reach out to his Congressional and/or Senatorial representatives' offices. Tell him you're intent to recognize one of their constituents. Politicians can "pipeline" such things.
3.) Regulations for awards area applicable to active duty members. How many our own are being recognized ex-post-facto? A lot of recent Medal of Honor recipients have been decorated well after being handed their DD-214.
4.) Have your write-up and summary done before you start knocking doors down.

If there is a will, there is a way. I applaud you for trying to get this Devildog some recognition.

Here is the excerpt right out of the Marine Corps Awards Manual (SECNAVINST 1650.1H):

c. Time limits are three years for submitting most
personal military decorations and five years for awarding them.
Exceptions to this are the Distinguished Flying Cross, with time
limits of two years for submission and three years for awarding,
and the Purple Heart Medal and Combat Action Ribbon, which do
not have submission time limits. Awards may be considered
beyond these time limits under the following circumstances:

(1) If SECNAV determines an individual's superior
submitted an appropriately documented award recommendation
within the specified time limits, but the recommendation was not
acted upon due to inadvertent loss during processing.

I don't know about the Navy specifically, but according to regs any person can recommend anyone for an award who has knowledge of the event. I know for things like the Combat action Badge and certain valor awards there are sworn statements and other supporting documents. The hard part may be finding people to provide those statements, hopefully officers or NCOs that are still active if possible.