Thanks. The listed version of Perl is 5.24.1

So that's an inconsistency. In Perl 5.24.1, '.' is in @INC, but your output does not show it. So someone has done something to remove it (and in fact, removing it manually was recommended in perl5421delta).

...and they insist they haven't changed anything.

For obvious reasons, I can not verify that claim. That's between you and them.

Excuse my ignorance but how can I discover "as the web server sees it."?

Since you haven't provided much details about your environment, I can only guess that this is some web hosting, and you can write CGI programs in Perl. If that's the case, write a CGI program which examines the directories in question and prints the file names, ownership and permissions, and also the user and group id under which the program is running.