Same problem on my 8320. Tried to download direct from my BB, then tried downloading through my computer. Get the message: "Please wait while BlackBerry App World is loaded on your device...." but nothing happens. Too bad this isn't as slick as iTunes store!
I went back to the Blackberry.com site via my 8320 just now and clicked to the download link. It worked just now, when it wouldn't work yesterday. I now have the Apps Store on my BB. Apparently the page was redesigned; it has a completely different appearance on my 8320 than it did yesterday. Good luck all!