Though I haven't found the perfect solution, I did find a couple of things. I haven't tested them yet but will soon.
I found a plugin called "VoucherPress" that will allow you create "coupons" and has the option to allow one coupon per member as well as entering your own keys. So you could set it that only one coupon code/key is delivered and set as expired, so the next person to claim a coupon won't get the same code/key.
The only drawback is that it creates the coupon as a PDF file. So your users will have to verify the coupon claim through email and then get linked back to the site's PDF file. The PDF files can be large and will be stored on your server, so after time it will eat up your space.
The only other option I've seen, and though it's not done through your site/wordpress, it's probably the better option so far.
You can create a Facebook "Fan Page" for your site and then add a WooBox.com Facebook APP that will allow automated game code key distribution. The free version of the APP allows 50 keys to be delivered per month. You can upgrade to the pay model to deliver unlimited keys per month.
If I can't find a good delivery system to use on my site, I'll probably use the Facebook method and pay for a months service to Woobox for the unlimited delivery, then downgrade back to the Free version when my codes have been distributed.
Good luck and if you find anything else that does this, let me know! :)