To elaborate on Spiff's answer, since it wouldn't easily fit in a comment:
Firefox ESR seems to work with java applet if it's 32-bit, and as long as you can keep it on v 52.x and disallow the auto updater from working. Download here.
After installing it, download 32 bit java (the non "64 bit" offline option--possibly remove any previous 64 bit java first, if necessary, to avoid a "1603 install failed" message, though sometimes the two"just work" together). Restart firefox after installing java.
Whenever there's a prompt for "firefox software updater" click "no" otherwise it might update it to 59. It'll allow you to load the plugin on pages now, but it's an "opt in, allow for this particular web page at a time".
Now test if applets are working here: https://www.java.com/en/download/installed.jsp
If you go with IE it works and isn't an opt-in (i.e. works "even better")--seems to require 32-bit JRE otherwise it just pops up a message "this website uses java" but despite saying OK, it still doesn't take, weird.
Another option: installed firefox portable ESR 52 version. By default it launches in either 64 or 32 bit mode "depending on the OS" (we want it to always launch in 32 bit mode), so to enforce that, copy "the FirefoxPortable.ini configuration file from the FirefoxPortable\Other\Source directory to the FirefoxPortable directory and editing it to add AlwaysUse32Bit=true" (thanks to grawity for the tip)
You may be able to use an "appletviewer" debug tool.