запуск Java-апплета в Firefox

Is there anyway to manually install the plugin?

No. You need to use 32-bit Firefox and a 32-bit JRE.

Java is not supported in 64-bit Firefox.

Firefox 64-bit for Windows, by design, has limited support for plugins and users will notice that certain sites requiring plugins that worked in previous 32-bit versions of Firefox might not work in this 64-bit version.

As we’ve shared previously, Mozilla intends to remove support for most NPAPI plugins in Firefox by the end of 2016.

Source Firefox 64-bit for Windows Available

You have the 64-bit version of Firefox, where only Silverlight and Flash Plugins are 'whitelisted'. If you need Java you need to install the 32-bit version of Firefox. You can have both the 64-bit and 32-bit versions installed, but you'll need to do a "custom" installation (on the first screen of the installer routine) and select or create a uniquely named folder - like maybe Firefox 32, otherwise the 32-bit installer may overwrite your 64-bit Firefox installation.

Source No Java plugin