I just tried the store, and I got something to download, using Firefox. (Chrome and Safari don't get along well with the store.) (And also, I'm on a Mac, because TS3 isn't uncooperative enough already.) But I did have a bit of trouble convincing the download to start. I opened my launcher first, which is usually necessary these days, but when Firefox asked me to select an "app" to open the download, my pack icons didn't work. The base game icon did, though, so try switching which one you use to download.
Also, as usual, clear your browser caches, including the one for Internet Explorer. The store and launcher use it even if you don't.
