Is there any web browser with NPAPI plugins support?

EDIT:

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

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.

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
    

In all honesty, we don’t have any good options today

Pale Moon has only partial support for Java Plugins. At least, they don’t work under Wayland. It is true for every browser, derived from Firefox, which for now is any browser from “NPAPI support list”. But at least Pale Moon team supports their project.

Gnome Web shares the same problem, but even to a higher degree. Same with Konqueror.

Midori used to be a good browser, but hasn’t been update for way too long

SeaMonkey doesn’t support NPAPI at all. They even state it on their website:

we are not planning to support any abandoned stuff like classic
extensions and NPAPI plugins on our own … when they are gone, they are gone

Safari has been gone for years now

So, yeah… I keep a virtual machine with Windows and (omg, this is embarrassing) IE installed, just to be able to work with my bank account

You have two options by the moment.

Basilisk

http://basilisk-browser.org/

Waterfox

https://www.waterfoxproject.org/

This answer is basically a compilation of all the good pieces of advice given in the other answers, but with the exhaustive steps to follow. And it has not been straightforward to me at all…

1. Download Java Development Kit SE 8 release 261 here for your system (should be Linux x64 Compressed Archive for Ubuntu nowadays standards), the last one to have Java plug-in support (as explained in this link of the second edit of the question).
2. Unpack it to /usr/lib/jvm/, using tar zxvf command properly.
3. Add it to the "Java alternatives" : sudo update-alternatives --install /usr/bin/java java /usr/lib/jvm/jdk1.8.0_261/bin/java 1 (the last number is for the priorities between the different Java versions co-existing in your OS. I say it doesn’t really matter since this is related to automatic installation and you should do everything manually here, and know what you’re doing…)
4. Select this freshly-installed alternative : sudo update-alternatives --config java
5. Run java -version (and not --version) and you should have some output like

java version "1.8.0_261"
Java(TM) SE Runtime Environment (build 1.8.0_261-b12)
Java HotSpot(TM) 64-Bit Server VM (build 25.261-b12, mixed mode)

6. Now you can install the last packaged version of Pale Moon for Ubuntu (which is actually an old one but who cares, since we’re trying to use a deprecated and unsecure technology anyway?) running the following commands:

echo 'deb http://download.opensuse.org/repositories/home:/stevenpusser/xUbuntu_22.04/ /' | sudo tee /etc/apt/sources.list.d/home:stevenpusser.list
curl -fsSL https://download.opensuse.org/repositories/home:stevenpusser/xUbuntu_22.04/Release.key | gpg --dearmor | sudo tee /etc/apt/trusted.gpg.d/home_stevenpusser.gpg > /dev/null
sudo apt update
sudo apt install palemoon

7. Before running Pale Moon, you still have to add the Java plug-in to the Firefox correct folder (so I guess you should have Firefox already installed in your OS), as explained here: sudo ln -s /usr/lib/java/jdk1.8.0_121/jre/lib/amd64/libnpjp2.so /usr/lib/mozilla/plugins/libnpjp2.so
8. Finally, run Pale Moon with your website (the good old official test page now actually recommands you to uninstall the plug-in, so you basically just have to pray…) and it should work!