There are apps - I use Grocery Gadget which has a web interface, as well as iOS, Android, Blackberry, Windows Phone, and other apps. I'm not thrilled with the web interface, but the iOS implementation is really nice.
Any of the generic list management web applications could work as a shopping list - RememberTheMilk, TeauxDeaux, Wunderlist, and so on. They have varying (but increasing) support for multiple platforms, poke at them until you find one you like.
Evernote could work, too. It syncs very well between devices (I use Windows, web, iOS, Android, additional platforms supported) and has a checkbox widget you can use to make lists that learn. You could use a single note as a shopping list, or have separate notes by store, or person. Sharing notes between multiple users works well.
For another approach using a mix of computer and paper, see Get It Done Guy on managing a shopping list