Ofc in this case feldrim above pointed us to the mac/linux/etc downloads so the estonian government has actually heard there are other platforms besides Windows.
I don't know about Estonia in particular but I'm guessing "$SOME_GOVERNMENT's interfaces" for most places is going to be HTTPS.
So, with that in mind, I've used a Nitrokey HSM and a Taglio PIVKey with Firefox on Linux using the OpenSC tools PKCS 11 module. I would suspect any smart card supported by OpenSC will work fine in Firefox.
From my reading, OpenSC is being distributed by the government of Estonia, so I suspect using it in a browser that supports PKCS 11 modules compatible with OpenSC on a Linux PC would work fine.
The app is used for changing PINs and there's another one for signing documents.
Signing in the browser uses a extension, code here https://github.com/open-eid/chrome-token-signing