If the site requires the Java plugin, there's no workaround. Please urge the site to update to standard HTML5, CSS, and JavaScript!

Meanwhile, Internet Explorer 11 can still run Java applets through an ActiveX control.

Just to be certain, are you sure it needs the Java Plugin from Oracle and is not JavaScript?

http://kb.mozillazine.org/JavaScript_is_not_Java

Firefox 52.0 and later Releases including 57.0.4 only allows the Flash Player Plugin Adobe to run.

Firefox 52 ESR allows other NPAPI Plugins to run. On Windows the Win64 only allows Flash Player and Silverlight to run so you would need 32-bit Firefox 52 ESR and 32-bit Java to work. https://support.mozilla.org/en-US/kb/npapi-plugins

You can check the Web Console (Tools -> Web Developer) for messages about blocked content and about the presence of mixed or unsafe content.

• https://developer.mozilla.org/en/Tools/Web_Console

You can check the Network Monitor to see if content is blocked or otherwise fails to load.

• https://developer.mozilla.org/en/Tools/Network_Monitor

If necessary use "Ctrl+F5" or "Ctrl+Shift+R" (Mac: Command+Shift+R) to reload the page and bypass the cache to generate a fresh log.

If you use extensions ("3-bar" menu button or Tools -> Add-ons -> Extensions) that can block (wanted) content (e.g. Adblock Plus, NoScript, Ghostery, uBlock Origin) always make sure such extensions do not block (wanted) content.

• make sure your extensions and filters are updated to the latest version
• https://support.mozilla.org/en-US/kb/troubleshoot-extensions-themes-to-fix-problems