I'd like to know if there is a way to determine the BIOS version of a given computer without rebooting and entering the BIOS at boot. In other words, is there some utility within XP that will display the BIOS version?

The BIOS information can easily be retrieved using a WMI script. Also, the same script can be run against a list of computers in your network and generate the output from the machine where you sit.

Here is a sample script that gathers the BIOS information from the local system. (Watch for line wraps)

strComputer ="."
Set objWMI = GetObject("winmgmts:\\" & strComputer & "\root\CIMV2")
Set colItems = objWMI.ExecQuery("SELECT * FROM Win32_BIOS")
For Each itm In colItems
      strBIOSVersion = Join(itm.BIOSVersion, ",")
      WScript.Echo "BIOSVersion: " & strBIOSVersion
      WScript.Echo "BuildNumber: " & itm.BuildNumber
      WScript.Echo "SMBIOSBIOSVersion: " & itm.SMBIOSBIOSVersion
      WScript.Echo "SMBIOSMajorVersion: " & itm.SMBIOSMajorVersion
      WScript.Echo "SMBIOSMinorVersion: " & itm.SMBIOSMinorVersion
      WScript.Echo "Version: " & itm.Version
Next

Additional resources

Microsoft Windows 2000 Scripting Guide - Retrieving Information About the BIOS

Microsoft TechNet: Scriptomatic Tool

Scriptomatic 2.0: Readme