Is there a web site for browsing or searching Apple's entire iTunes Store catalog, without having to install and use iTunes software?
There's no browsing, but you can search via Marco Arment's Preview.fm. It's very fast. (Music only.)
You can browse the top 100 Songs, Albums, TV Shows, Movies, Movie Rentals, Free Apps, Paid Apps and Music Videos online:
Of course, if you want to buy any of them you will have to install iTunes.
If you just want to look through stuff I found this that may or may not help. Looks a little sketchy to me but that's probably just the design of it. Plus it doesn't look like there's a search feature.
You can preview some things on the itunes.apple.com website - like this podcast listing for Tech News Today... but most stuff you need to have iTunes installed.
I use AppShopper for iOS apps. It's very extensive and updated extremely quickly.
It's not an exact replica of the search functionality, but Apple's website serves iTMS pages to your browser--try searching apple.com for an artist or app. These pages usually force iTunes to open automatically and navigate to the equivalent page in the store, but you can halt that behavior by following the instructions here: http://theappleblog.com/2010/03/15/how-to-stop-itunes-web-links-from-opening-itunes/