Is it possible to upgrade Windows Server 2008 32bit (Web Server) to 64 bit, or would it require a clean install?
It would require a fresh install.
Windows has never supported a x86/32-bit to a x64/64-bit upgrade in any operating system.
Ask yourself, would you WANT to play the upgrade game with a MS product like the OS? I can't recall anytime that an upgrade was perfect.
In the old days, of 95-98 upgrades, it would leave bulk behind, or old files, data / garbage.
With a server, I wouldn't play games, reformat, start fresh with a clean slate.