If you look in your badges tab and set the discriminator thingie to "Veteran", you should see both the badges themselves and things that look like the badges that will have little buttons on them that say "Claim Reward" or something like that. Click the button--if there's an additional choice (i.e. sands of mu or ghost slaying axe) a menu will pop up for you to choose from.
Open the pod bay doors, Hal.
...!
Ah, there we go. Wow. I never even looked at that before. XD; Thanks!
Global: @Kenka Banchou
Server: Virtue
Twitter: @mugennoken
We were all new once. 
I remember asking support how I do I claim vet rewards. Worryingly the first person couldn't answer it and escalated it to a 'senior customer support' dude. 
SAVE CoX info:
Titan Network efforts
Saving CoX events/FB info
I was having trouble finding these as well, and also felt like a total noob. At least I wasn't alone. 
Well, I certainly haven't felt like such a newb in awhile, but here goes.
I just hit my first year anniversary for CoH, got my shiny new 12 month badge, and all that fun stuff.
Truth be told, I haven't paid much mind to when I've gotten my previous rewards, just that they have come into effect a few weeks after I realized I'd hit them. However, besides the badge, I haven't gotten any sort of notification on the Reward powers, nor any way to select anything, etc...
Is there a certain amount of time after the anniversary date that they go into effect, are they supposed to be at the same time as the badge, or what? >_>
Just curious.
Global: @Kenka Banchou
Server: Virtue
Twitter: @mugennoken