My landlord wants a copy of my orders. Does he get them?

As a landlord myself. The only reason he should need orders is if you where using the service member civil relief act to break the lease due to military service.

If your lease is up at the same time as your PCS there is no need to prove why you are moving. Thank you for your service.

I don't understand the reason for your landlord's request. If you're not breaking the lease what difference does it make where you are going after your lease expires. Could it possibly be that you did not give sufficient time period advance move out notification to landlord or something like that and you could be penalized legally/financially for not making a timely notification prior to move out at lease expiration time?
It doesn't matter if he is a military officer reservist as far a "trusting" him or not goes The only thing that matters is whatever the law states regarding these matters.
I would not give him a copy unless a military legal assistance officer (lawyer) tells you that you are required by law to do it to avoid any legal/ financial penalties on you I would immediately contact the nearest U.S. military legal assistance office for advice/guidance (it doesn't matter what Branch of Service)