Is there any web browser with NPAPI plugins support?

The Pale Moon web browser is an interesting option. In the document Pale Moon future roadmap they make this statement:

"Pale Moon supports NPAPI plug-ins. Unlike Firefox, we will not be deprecating or removing support for these kinds of plug-ins. This means that you will be able to continue using your media, authentication, and other plug-ins in Pale Moon like Flash, Silverlight, bank-authenticators or networking plug-ins for specific purposes."

You can use the Extended Support Release of Firefox 52 as follows:

The following steps should work (tested on Mint 18 based on Ubuntu 16.04 and Mint 19 based on Ubuntu 18.04):

1. Download and extract the Firefox ESR install from here: Download Firefox Extended Support Release. Make sure it doesn't auto update by changing the update settings in preferences.
2. Launch Firefox and type about:config in the location bar
3. Accept the risks and then add a key (right click mouse -> New -> Boolean)
4. Call the key plugin.load_flash_only and set it to false
5. Add a Firefox extension for switching the user agent eg. User-Agent Switcher
6. Set the user agent to an earlier version of Firefox on Linux eg. Mozilla/5.0 (X11; Linux i586; rv:31.0) Gecko/20100101 Firefox/31.0
7. Close Firefox
8. Download the tar.gz JRE package for Linux from Java.com and store the file in the Downloads folder in your home folder (i.e. ~/Downloads). It will be of the form: jre-8u121-linux-i586.tar.gz for the 32 bit version. Choose the 32 bit or 64 bit version according to the version of Firefox ESR you downloaded.

9. Then do the following in a shell to extract the JRE (replacing with 64 bit versions if needed):

   sudo mkdir -p -v /opt/java/32
   cd ~/Downloads
   tar -zxvf jre-8u121-linux-i586.tar.gz
   sudo mv -v jre1.8.0_* /opt/java/32
   

10. Now the Java needs to be linked to Firefox. In a shell:

    mkdir -p ~/.mozilla/plugins/
    cd /opt/java/32/jre1.8.0_121
    ln -sf $PWD/lib/i386/libnpjp2.so ~/.mozilla/plugins/
    

11. Launch Firefox and verify your Java version here: Verify Java Version

12. If it doesn't work and the wrong Java plugin appears to be loaded eg. IcedTea, then remove them:

    sudo apt-get remove icedtea-plugin icedtea-8-plugin icedtea-netx icedtea-netx-common
    

EDIT:

Here's a list of browsers that still support NPAPI from Wikipedia:

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There is a special version of Firefox that still supports NPAPI plugins.

From the How-To-Geek:

Mozilla ended support for traditional NPAPI browser plugins, aside from Flash, with Firefox 52 on March 7, 2017.

However, Mozilla offers an “Extended Support Release”, or ESR, branch of the Firefox browser. This browser is intended for organizations to have a stable, long-term platform that only receives security updates, not the frequent feature updates and changes that the main version of Firefox receives.

Firefox 52 ESR was released on March 7, 2017 and includes support for non-Flash browser plugins. Mozilla will continue supporting Firefox 52 ESR with security updates until sometime in the second quarter of 2018. At that point, the next ESR version of Firefox will drop support for NPAPI plugins.

Visit the Download Firefox Extended Support Release page to download the ESR version of Firefox. If you’re not sure which version to download, choose the 32-bit version of Firefox for maximum compatibility with older plugins. Older plugins may not have 64-bit versions available.