There are two meanings of "admin": the one who is given a role administrator in the MediaWiki website, and the one who controls database and the server. I don't know which one were you referring to (it is also possible that both are the same person).
Assuming that $wgAllowUserCss = false;
Then, if you managed to find someone who can access the server, you could ask that person to edit LocalSettings.php so that $wgAllowUserCss = true;, and now your User:Chuck/Print.css will work. Note that it is absolutely impossible to make your User:Chuck/Print.css work while $wgAllowUserCss = false;.
Otherwise, if you managed to find someone who is given a role administrator, and it is okay to make your change to Print.css systemwide, you can ask the administrator to edit MediaWiki:Print.css. This guarantees to work even when $wgAllowUserCss = false;.
Usually there could be more than one user who are given the role administrator. You could see the list of administrators by going to the page Special:ListUsers. For example, here's all administrators in English Wikipedia: https://en.wikipedia.org/w/index.php?title=Special%3AListUsers&username=&group=sysop&limit=50
Please also note that if you can access the database, you can give a role administrator to a user in the website via the database too.
Assuming that $wgAllowUserCss = true;
It is also possible that in fact $wgAllowUserCss = true;, but the change doesn't take effect. Read https://en.wikipedia.org/wiki/Wikipedia:Bypass_your_cache for the solution.
Note: without an admin (in both meanings), pages cannot be deleted, and a lot of system changes cannot be done. Your company wouldn't be able to stay in this state forever. Perhaps you should ask your company to install a new wiki.