We have a set of web servers running the standard version of Windows 2008.
We need to upgrade the Enterprise edition due to high availability functionality.
Can we just run an upgrade of each of the web servers? What will happen to our existing applications, will they need to be reinstalled?
Yes, upgrading from Server 2008 Standard to Server 2008 Enterprise is a supported upgrade path: Windows Server 2008 R2 Upgrade Paths. The applications should not have to be reinstalled but as with any upgrade you should make sure you have a good backup before proceeding.
Yes it should work fine. Boot into your Windows Server 2008 Standard, insert the Windows Server 2008 disk and run it. Input a Windows Server 2008 Enterprise key or choose Windows Server 2008 Enterprise and then you can perform the upgrade installation.
http://technet.microsoft.com/en-us/library/cc755199(WS.10).aspx#BKMK_Supported