How about simply, Besha'ah Tovah!
Following the lead of the new fruit for the second day of Rosh Hashana (because we're not sure if we should say that bracha for that day), shouldn't you find some other reason to say Shehecheyanu in close proximity to the questionable event?
I think the question about the website being a physical entity that one finishes or acquires is an interesting one - can it get a brocha? Like a new shirt of a new tallis?
I think yes. Any website still has existence on a server somewhere - even if its data is stored energy on a laser disk - it still exists as a physical recording on the physical plane.
Now I think the more interesting question - is that do we need to stand next to the designated server in order to make the brocha - or is looking at the computer screen enough? And by extension can I make a brocha on a picture of a new tallis that someone bought for me and took a photo of?
hatov vehametiv I'm certain that you don't, the only case is when someone brings a second bottle of wine during a meal you have with other people.
as for sheehianu, it would make sense if you are happy, I guess comparable to printing a book. but the website is not done, it is a work in progress and also there's no takana of beracha on this case.
it doesn't mean that you shouldn't thank hashem for it, praying or making a seuda.