Is there a way to have a script populate a field with the date or time from the server clock rather than the client computer clock? If not, does anyone know of extensions that do this?

Best bet is to synchronise both the host and client machines to the *same* network time server. Software for this is built into MacOS, probably built into Windows too. Or it will be easy to find.

I tinkered around with the Server Administration external plug-in.

There are some date/time functions in there