What is the process (in the Army) for someone wanting to concealed carry on post?

Zero chance to carry on post. The Order is intended to provide self protection for military personnel who are not on a post or base with military Police presence. Think Recruiters, Reserve drilling locations. If the post has MP's or civilian police then then you don't meet the criteria. There is a Federal Directive against concealed carry on base with very, very few exemptions. You need to read the entire order not just the "authorization" part. Semper Fi.

SSG James Elmore "What is the process (in the Army) for someone wanting to concealed carry on post?"
Let me start with.... Sgt, are you emotionally attached to both testicles? because likely it will take the swap of one or both before your going to get a post commander to do what he "can" do under that reg.

And as pointed out, the DOD directive states in part

"1.2. POLICY.
a. When authorized, DoD personnel may be armed on and off DoD property when required
for the performance of official duties.
b. Arming DoD personnel other than law enforcement and security personnel will be
considered for locations where law enforcement or security personnel are not located on site or
in a reasonable proximity"

So on Fort Wainwright would not apply.

For those interested in facts not what they heard..http://www.dtic.mil/whs/directives/corres/pdf/521056_dodd_2016.pdf

So, my understanding locally is that at the VA facilities (Not an official military installation) with a CCW concealed carry is permitted. When I first started going to the VA I made a point to stop by the VA Police office here and present my CCW permit and ask if carry was allowed. I was given the green light. However, I would not even think about taking a firearm onto a military installation.