Hey Crackers- I'm as baffled as you are regarding desktop iTunes. I get the gift option for and music/tv/movies I own, but not for the apps! I tried every option and workaround I could think of within the store (send from purchased list? nope. vis send gifts on home page? nope. send via app's weblink? nope!!!!)
But... you CAN logout of your itunes account, navigate to the app you want to send, click send as gift, and login when requested. It ridiculous, but works.
FYI:
itunes 11.0.2
iOS 6.1
As for the lack of gift buttons, I can confirm that my ipad doesn't show the button like my phone does. It is also running 6.1, but the OS is shown like this:
Version 6.1 (10B141)
whereas my phone says:
Version 6.1 (10B143)
I'm not sure if the discrepancy is iphone to ipad or something else. Regardless, I though you might like as much info as possible!
-ash