Unfortunately, Minecraft: Story Mode never released officially on Linux, meaning there is no native version and there's no way to get it legitimately. Although, it is quite easy to get the Windows version running on any Linux distro. If you're a Linux enthusiast, you'll probably already know what I'm talking about, but for all the Linux newbies, We'll provide a small tutorial.
Lutris
First, download the game from your preferred Windows source.
If you don’t already have it, install Lutris.
Once you have Lutris open, click the + icon in the top left corner, then click “Install a Windows game from an executable.”
Type “Minecraft: Story Mode” as the name (or whatever you like), then click “Install.”
Next to Wine, click “Install” again.
On the next page, change any settings you like, but it’s recommended to leave it as default. You can also choose if you’d like to create a desktop or app menu shortcut.
Once you’re done, click “Continue.”
On the next page, click “Browse” and choose the setup file from where you extracted the Windows download, then click “Install.”
Once it does it’s thing, it’ll open up the installer like any old Windows machine. Run through the installer like normal. Once it’s done you should be able to launch the game through Lutris!
IMPORTANT!!!: If you’re installing the GOG version, you’ll need to install the DLC episodes as well. To do this, in Lutris, simply select Minecraft: Story Mode, and on the bottom bar where it has the “Play” button, click the drop-down button next to the button that looks like a painting, and click “Run EXE inside Wine prefix,” and select the Adventure Pass installer from your extracted download folder. Run through the installer like normal, and all the episodes should be installed. You do NOT have to do this for Season Two.