I couldn't find an Apple site where you can search the iTunes store online.

That said, many other features are available online from Apple, and there are third-party search options.

You can access top charts and featured stuff here: https://www.apple.com/itunes/charts/ (Trying to click the Shop the iTunes Store now button on that page just opens iTunes, which can't access the iOS App Store anymore.)

I accidentally stumbled upon this page with the Categories, so you can browse apps by category alphabetically or by what's popular: https://itunes.apple.com/us/genre/ios/id36?mt=8. This looks like an old page, and the interface isn't great.

Some unofficial search workarounds might be Googling the app name and adding site:itunes.apple.com/us/app/ to your query, or using http://appshopper.com/.

Hope this helps.